Ponnusamy Sugawa, Vuorinen. (Eds.) Quasiconformal Mappings and Their Applications. IWQCMA05 (Draft, Madras, 2006)(ISBN 8173198071)(358s)_MCc

International Workshopon

QUASICONFORMAL MAPPINGS AND THEIR APPLICATIONS

December 27, 2005 - January 01, 2006

Indian Institute of Technology Madras

Department of Mathematics

Proceedings of the

International Workshopon

Quasiconformal Mappings And Their Applications

(IWQCMA05)

December 27, 2005 - January 01, 2006

Edited by

S. PonnusamyT. Sugawa

M. Vuorinen

Co-organized by

• Chennai Mathematical Institute, Chennai, India

• Institute of Mathematical Sciences, Chennai, India

Sponsored mainly by

• National Board for Higher Mathematics (DAE),India

• Forum d’Analystes, Chennai, India

• National Science Foundation, USA

• The Abdus Salam International Center for Theoretical Physics,Italy

• Commission on Development and Exchanges of the InternationalMathematical Union

• Indian National Science Academy, India

• Council of Scientific and Industrial Research, India

• Department of Science and Technology, India

Preface

The Department of Mathematics, IIT Madras, Chennai hosted InternationalWorkshop on Quasiconformal Mappings and their Applications (IWQCMA05)December 27, 2005- January 1, 2006. This event was the first one in India onthis active research area which has its roots in geometric function theory andwhich is closely connected with several topics of mathematical analysis.

The organizers gratefully acknowledge the financial support of

1. National Board for Higher Mathematics (DAE), India2. National Science Foundation, USA3. The Abdus Salam International Center for Theoretical Physics, Italy4. Commission on Development and Exchanges of the International Mathe-

matical Union, Italy5. Indian National Science Academy, India6. Council of Scientific and Industrial Research, India7. Department of Science and Technology, India8. Forum d’Analystes, Chennai, India.

We also thank

1. Theivanai Ammal College for Women, Viluppuram, India2. Canara Bank, IIT Madras Campus3. State Bank of India, IIT Madras Campus

for their support. Many people have given us help, in particular the students ofProf. S.Ponnusamy. Prior to the start of the workshop, preworkshop lectureswere given by Dr. Antti Rasila and Prof. Raimo Nakki from Finland. We thankboth of them. Also, we take this opportunity to thank Prof. R. Balasubramanian,Prof. R. Parvatham, and Prof. C. S. Seshadri for their continued encouragementand helpful advice.

The participants, who represented many different countries, received in mostcases financial support from their national funding organizations to cover theirexpenses. ICTP’s generous support was useful in supporting mathematiciansfrom developing countries. Without the invaluable support from the aforemen-tioned organizations, this conference would not have been possible in its presentform.

The main goal of the conference was to bring together internationally well-known experts representing geometric function theory and some related topics.They were requested to deliver a series of lectures for postgraduate students ontheir respective areas. The audience consisted of mathematicians ranging fromgraduate students to well-known experts from all the participating countries.

Conformal invariance and conformally invariant metrics have been importantresearch topics in geometric function theory during the past century. Thesetopics also were discussed or mentioned in several of the lectures. The organizingcommittee was pleased to observe that the lectures were very well received and

lead to many lively discussions afterwards. We were also pleased to receivepositive response from the speakers to our request to contribute their lecturesfor the proceedings. It is our hope that the publication of these proceedings willmake the results presented in this Workshop and also this research area and itschallenging open problems more widely known for a wide readership than whatis the case presently. The editorial work was carried out at IIT Madras, and thewww-pages

http://mat.iitm.ac.in/ samy/

http://www.cajpn.org/madras/

http://www.math.utu.fi/proceedings/madras

contain a copy of these proceedings.

Special thanks to Mr. Swadesh Kumar Sahoo, and Mrs. P. Vasundhra fortheir help in the organization of the meeting. We take this opportunity to thankProf. Roger W. Barnard for his support in getting NSF grant for supporting USparticipants, and for its partial support in bringing out this proceedings.

On behalf of the Organizing committee

S. PonnusamyIIT Madras

T. SugawaHiroshima University

M. VuorinenUniversity of Turku

ContentsPreface

Roger W. Barnard, Clint Richardson, 1Alex Yu. SolyninA note on a minimum area problem fornon-vanishing functions

Alan F. Beardon and David Minda 9The hyperbolic metric and geometric function theory

Peter Hasto 57Isometries of relative metrics

David A Herron 79Uniform spaces and Gromov hyperbolicity

Ilkka Holopainen, and Pekka Pankka 117p-Laplace operator, quasiregular mappings,and Picard-type theorems

Henri Linden 151Hyperbolic-type metrics

Williams Ma and David Minda 165Geometric properties of hyperbolic geodesics

Olli Martio 189Quasiminimizers and potential theory

R. Michael Porter 207History and recent developments in techniques fornumerical conformal mapping

Antti Rasila 239Introduction to quasiconformal mappings in n-space

Toshiyuki Sugawa 261The universal Teichmuller space and related topics

Matti Vuorinen 291Metrics and quasiregular mappings

G. Brock Williams 327Circle packing, quasiconformal mappings,and applications

List of Participants 349

Proceedings of the International Workshop on QuasiconformalMappings and their Applications (IWQCMA05)

A Note on a Minimum Area Problem for Non-VanishingFunctions

Roger W. Barnard, Clint Richardson, Alex Yu. Solynin

Abstract. We find the minimal area covered by the image of the unit diskfor nonvanishing univalent functions normalized by the conditions f(0) =1, f ′(0) = α. We discuss two different approaches, each of which contributesto the complete solution of the problem. The first approach reduces the prob-lem, via symmetrization, to the class of typically real functions, where we canemploy the well known integral representation to obtain the solution uponprior knowledge about the extremal function. The second approach, requiringsmoothness assumptions, leads, via some variational formulas, to a boundaryvalue problem for analytic functions, which admits an explicit solution.

Keywords. Symmetrization, Minimal Area Problem.

2000 MSC. 30C70.

Contents

1. Introduction 1

2. Outline of Our Method 4

3. The Iceberg Problem 6

References 8

1. Introduction

Let D = z : |z| < 1 and Ap =

f analytic in D :

∫

D

|f(z)|p dA = ||f ||pAp < ∞

,

the Bergman space of analytic functions in D.

Recently, Aharanov, Beneteau, Khavinson, and Shapiro [2] considered a gen-eral minimization problem on A

p

inf||f ||Ap : f ∈ Ap, ℓi(f) = ci, i = 1, . . . , n

where ℓi are bounded linear functionals on Ap, p > 1. They proved several general

results about this problem.

Version October 19, 2006.Supported by NSF grant DMS-0412908.

2 Roger W. Barnard, Clint Richardson, Alex Yu. Solynin IWQCMA05

As we know, in recent years tremendous progress has been achieved in thestudy of Bergman spaces. For a detailed account of this progress, we refer to therecent monograph by Peter Duren and Alex Schuster, Bergman Spaces, [7].

Aharanov, Beneteau, Khavinson, and Shapiro [2] also mentioned that to ob-tain a complete solution of a particular problem, one often needs additionalinformation which does not follow from their methods.

A particular example is the following open problem:

inf

∫

D

|f |2 dA : f 6= 0 in D, f(0) = 1, f ′(0) = α

This is a “typical” extremal problem on the class of non-vanishing analyticfunctions. The nonlinearity of the class is the obvious obstacle here.

But, we have a method which allows us to solve some problems similar to thisone.

Let

Nα = f : f is univalent, and non-vanishing on D,

f(z) = 1 + a1(f)z + . . . ,

normalized by a1(f) = α The area of the image f(D) is given by

D(f) =

∫

D

|f ′|2 dA = π

∞∑

n=1

n|an(f)|2.

ThusD(f) ≥ πα

2,

with equality iff f(z) = 1 + αz.

Since this map f is in Nα, 0 < α ≤ 1, Koebe’s 1/4 Theorem implies Nα = ∅for α > 4. So the nontrivial range is 1 < α < 4 .

For the non-trivial range, the minimal area problem for Nα is solved by

Theorem 1.1. For 1 < α < 4, let f ∈ Nα. Then

D(f) ≥παa2

(a +

√a2 − 1

)2 (αa

2 − 2√

a2 − 1(a +

√a2 − 1

))(1.1)

where a = a(α) is the solution to

1

α= a

2

[1 −

√a2 − a(a +

√a2 − a)3 log

((a +

√a2 − 1)4

/16a2(a2 − 1))]

,(1.2)

which is unique in the interval 1 < α < ∞.

Equality in (1.1) holds iff f = fα defined by

fα(z) =

∫ z

−1

−β

√ξ2 − a2

(ξ +

√ξ2 + 1

)2 (a

√ξ2 − 1 + ξ

√a2 − 1

)dz

z

A Note on a Minimum Area Problem for Non-Vanishing Functions 3

1 2 3 4

Α

20

40

60

80

100

AHΑL

Α(α)

πα2

Figure 1. The graph of A(α).

4 8-4

4

-4

Α = 3

Figure 2. The extremal domain Dα = fα(D) for α = 3.

with ξ = ia2

1−z√z

and β = αa2(a +

√a2 + 1

).

For 0 < α < 4, let

A(α) = minf∈Nα

D(f)

denote the minimal area covered by the images of functions in the class Nα. NoteA(α) is convex and increasing. This can be proven from the formulas, geometry,and variational arguments. See Figure 1.


2. Outline of Our Method

First consider the minimal area problem on Tα, the typically real nonvanishingfunctions (not necessarily univalent). Use the linear structure of Tα and refor-mulate to show uniqueness and get simple “sufficient conditions” for extremalitycorresponding to linearized functions. This gives

Theorem 2.1. For 1 < α < 4, let f ∈ Tα. Then (1.1) holds with the same cases

of equality.

The technique of this proof was developed earlier in [1]. What is missing ishow to construct the extremal function!

Next. Assuming sufficient smoothness, we can apply a variant of Julia’s Vari-ational Formula in [5]. This leads to boundary conditions for an extremal analyticfunction. To obtain this “conditional” solution requires a priori smoothness.

Next, to achieve the “required smoothness,” we exploit geometric control ofthe mapping radius and apply standard symmetrization techniques to obtain thesufficient initial Jordan rectifiability as in [4].

Then we can apply earlier smoothing variations developed by Barnard andSolynin in [5] giving “required smoothness.”

Thus the “conditional” proof becomes a true proof. We then verify thatthe function recovered from the first step satisfies the sufficient conditions ofextremality which also leads then to a complete solution of the problem.

For a first step on Tα, we renormalize so that

f(0) = 1, f ′(0) = α.

Subordination implies 0 < α ≤ 4. Since Tα is compact and convex, the mini-mizer exists and is unique. The uniqueness follows by letting f1 and f2 be twominimizers. Then

D((f1 + f2)/2) =1

4

∫

D

|f ′1+ f

′2|2 dσ(2.1)

≤ 1

2

(∫

D

|f ′1|2 dσ +

∫

D

|f ′2|2 dσ

)

=1

2(D(f1) + D(f2)) ,

with equality iff f′1≡ f

′2.

We note here that the uniqueness obtained here is fortunate, since uniquenessis in general not obtained when variational and approximation methods are used.

Reformulating the problem using the linearity of Tα, we use the followinglemma from [1, 3]

Lemma 2.2. For f′α continuous on D, fα minimizes D(f) on Tα iff fα minimizes

L(f) = ℜ∫

D

f ′α(z)f ′(z) dσ

on Tα.


Proof. See Lemma 1 of [3].

Lemma 2.3. If f′α is continuous on D, then

L(f) =

∫ π

0

Kα(t)dµf (t),

where

Kα(t) =2πα

sin tℑ

e

itf′(eit)

.

Proof. See [3].

Proof of Theorem 2.1 for Tα. For Dα = fα(D), first show (0, 1] ⊂ Dα byconsidering

f(z) = 1 − 1

τ+

fα(τz)

τ

for τ < 1 and compare D(f) with D(fα). Then fα(−1) = 0 since fα is notidentically 0 and fα ∈ Tα.

Thus with Lemmas 2.2 and 2.3, fα minimizes D(f) on Tα iff fα minimizesL(f) under the constraints

2

∫ π

0

dµf = 1

∫ π

0

sec2

(t

2

)dµf =

2

α

Now we can use well known results to show fα is extremal iff Kα satisfies

Kα(t) = λ0 + λ1 sec2

(t

2

)∀ t ∈ Supp (µfα

)

Kα(t) ≥ λ0 + λ1 sec2

(t

2

)∀ t /∈ Supp (µfα

),

where λ0, λ1 are real constants.

Long computations, see [3], show our fα gives Kα that satisfies these condi-tions!

Next we characterize the geometry of extremal domains for Nα.

Lemma 2.4. 1. ∀α, 1 < α < 4, an extremal fα minimizing D(f) exists in

Nα.

2. If fα is extremal, then fα(D) is bounded, starlike with respect to 1, and

circularly symmetric with respect to rays

ℓτ = z = x + iy : y = 0, x ≥ τ,∀ τ, 0 ≤ τ ≤ 1.3. The min area A(α) := D(fα) for 1 < α < 4.

Proof. Apply circular and radial symmetrizations, then polarizations similar toarguments in [5].


4 8-4

4

-4

Α = 3

Lfr

nfL1

αf

Figure 3. The free (Lfr) and non-free (Lnf ) portions of the boundary.

Now combine Theorem 2.1 and Lemma 2.2 to see that if fα is extremal in Nα,then since fα ∈ Tα, Theorem 2.1 implies Theorem 1.1.

Next we show how the extremal fα in Nα can be recovered from its boundaryvalues.

Lemma 2.5. Let fα be extremal for Nα. Then f′ is continuous on D and |f ′| ≡

β ≥ α ∀ z ∈ ℓfr. See Figure 2.

Proof. Apply the deep “two point variation techniques” from [5] twice givingf′ these properties on ℓfr. Then use the Julia-Wolff Theorem and boundary

behavior properties from Pommerenke [6], giving f′ these properties everywhere.

Lemma 2.6. If fα is extremal, ϕ(z) = log(zf ′α(z)) maps as described in Figure 2,

with

q1(z) =i(1 − z)

2 sin(ϕ

0

2)√

z

ϕ2(ξ) = ci

∫ ξ

0

t2 − b

2

(t2 − a2)√

1 − t2dt + s.

Long computations are used to show monotonicity, then we use line integralformulae to compute the area as in [4].

3. The Iceberg Problem

A related problem is known as the Iceberg Problem: Given a fixed volumeabove the water, how deep can the iceberg go? See Figure 3.

This problem can be modeled by supposing a slice III is a continuum in C andE = III : cap III = 1, area [III ∩ UHP] = α.


0 τ α = −1

UHP

log |f ’(1)|

qϕ

21

ϕ

Figure 4. The mapping ϕ(z) = log(zf ′α(z)).

h ?

I

Figure 5. The Iceberg Problem.

We anticipate using similar arguments to those in this paper to find

h = minIII∈E

minℑz : z ∈ III.


References

[1] D. Aharonov, H. S. Shapiro, and A. Yu. Solynin, Minimal area problems for functions with

integral representation, J. Analyse Math., to appear.[2] Dov Aharonov, Catherine Beneteau, Dmitry Khavinson, and Harold Shapiro, Extremal

problems for nonvanishing functions in Bergman spaces, Selected topics in complex analysis,Oper. Theory Adv. Appl., vol. 158, Birkhauser, Basel, 2005, pp. 59–86.

[3] R. W. Barnard, C. Richardson, and A.Yu Solynin, A minimal area problem for non-

vanishing functions, Analysis and Algebra, accepted.[4] R.W. Barnard, C. Richardson, and A.Yu Solynin, Concentration of area in half planes,

Proc. Amer. Math. Soc. 133 (2005), no. 7, 2091–99.[5] R.W. Barnard and A.Yu Solynin, Local variations and minimal area problems, Indiana

Univ. Math. J. 53 (2004), no. 1, 135–167.[6] Ch. Pommerenke, Boundary behavior of conformal maps, Springer-Verlag, 1992.[7] Alex Schuster and Peter Duren, Bergman spaces of analytic function, Mathematical Surveys

and Monographs, no. 100, American Mathematical Society, Providence, RI, 2004.

Roger W. Barnard, Clint Richardson, Alex Yu. Solynin Address: Department of

Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409

Address: Department of Mathematics and Statistics, Stephen F. Austin University, Nacog-

doches, Texas 75962

Address: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas

79409

E-mail: [email protected]: [email protected]: [email protected]


The hyperbolic metric and geometric function theory

A.F. Beardon and D. Minda

Abstract. The goal is to present an introduction to the hyperbolic metricand various forms of the Schwarz-Pick Lemma. As a consequence we obtaina number of results in geometric function theory.

Keywords. hyperbolic metric, Schwarz-Pick Lemma, curvature, Ahlfors Lemma.

2000 MSC. Primary 30C99; Secondary 30F45, 47H09.

Contents

1. Introduction 10

2. The unit disk as the hyperbolic plane 11

3. The Schwarz-Pick Lemma 16

4. An extension of the Schwarz-Pick Lemma 19

5. Hyperbolic derivatives 21

6. The hyperbolic metric on simply connected regions 24

7. Examples of the hyperbolic metric 28

8. The Comparison Principle 33

9. Curvature and the Ahlfors Lemma 36

10. The hyperbolic metric on a hyperbolic region 41

11. Hyperbolic distortion 45

12. The hyperbolic metric on a doubly connected region 47

12.1. Hyperbolic metric on the punctured unit disk 47

12.2. Hyperbolic metric on an annulus 49

13. Rigidity theorems 51

14. Further reading 54

References 55

Version October 19, 2006.The second author was supported by a Taft Faculty Fellowship and wishes to thank the Uni-versity of Cambridge for its hospitality during his visit November 2004 - April, 2005.

10 Beardon and Minda IWQCMA05

1. Introduction

The authors are writing a book, The hyperbolic metric in complex analysis,that will include all of the material in this article and much more. The ma-terial presented here is a selection of topics from the book that relate to theSchwarz-Pick Lemma. Our goal is to develop the main parts of geometric func-tion theory by using the hyperbolic metric and other conformal metrics. Thispaper is intended to be both an introduction to the hyperbolic metric and a con-cise treatment of a few recent applications of the hyperbolic metric to geometricfunction theory. There is no attempt to present a comprehensive presentation ofthe material here; rather we present a selection of several topics and then offersuggestions for further reading.

The first part of the paper (Sections 2-5) studies holomorphic self-maps ofthe unit disk D by using the hyperbolic metric. The unit disk with the hyper-bolic metric and hyperbolic distance is presented as a model of the hyperbolicplane. Then Pick’s fundamental invariant formulation of the Schwarz Lemma ispresented. This is followed by various extensions of the Schwarz-Pick Lemmafor holomorphic self-maps of D, including a Schwarz-Pick Lemma for hyperbolicderivatives. The second part of the paper (Sections 6-9) is concerned with theinvestigation of holomorphic maps between simply connected proper subregionsof the complex plane C using the hyperbolic metric, as well as a study of neg-atively curved metrics on simply connected regions. Here ‘negatively curved’means metrics with curvature at most −1. The Riemann Mapping Theorem isused to transfer the hyperbolic metric to any simply connected region that isconformally equivalent to the unit disk. A version of the Schwarz-Pick Lemma isvalid for holomorphic maps between simply connected proper subregions of thecomplex plane C. The hyperbolic metric is explicitly determined for a numberof special simply connected regions and estimates are provided for general sim-ply connected regions. Then the important Ahlfors Lemma, which asserts themaximality of the hyperbolic metric among the family of metrics with curvatureat most −1, is established; it provides a vast generalization of the Schwarz-PickLemma. The representation of metrics with constant curvature −1 by boundedholomorphic functions is briefly mentioned. The third part (Sections 10-13) dealswith holomorphic maps between hyperbolic regions; that is, regions whose com-plement in the extended complex plane C∞ contains at least three points, andnegatively curved metrics on such regions. The Planar Uniformization Theoremis utilized to transfer the hyperbolic metric from the unit disk to hyperbolicregions. The Schwarz-Pick and Ahlfors Lemmas extend to this context. Thehyperbolic metric for punctured disks and annuli are explicitly calculated. Anew phenomenon, rigidity theorems, occurs for multiply connected regions; sev-eral examples of rigidity theorems are presented. The final section offers somesuggestions for further reading on topics not included in this article.

The hyperbolic metric and geometric function theory 11

2. The unit disk as the hyperbolic plane

We assume that the reader knows that the most general conformal automor-phism of the unit disk D onto itself is a Mobius map of the form

(2.1) z 7→ az + c

cz + a, a, c ∈ C, |a|2 − |c|2 = 1,

or of the equivalent form

(2.2) z 7→ eiθ z − a

1 − az, θ ∈ R, a ∈ D.

It is well known that these maps form a group A(D) under composition, and thatA(D) acts transitively on D (that is, for all z and w in D there is some g in A(D)such that g(z) = w). Also, A(D, 0), the subgroup of conformal automorphismsthat fix the origin, is the set of rotations of the complex plane about the origin.

The hyperbolic plane is the unit disk D with the hyperbolic metric

λD(z)|dz| =2 |dz|

1 − |z|2 .

This metric induces a hyperbolic distance dD(z, w) between two points z and w

in D in the following way. We join z to w by a smooth curve γ in D, and definethe hyperbolic length ℓD(γ) of γ by

ℓD(γ) =

∫

γ

λD(z) |dz|.

Finally, we setdD(z, w) = inf

γℓD(γ),

where the infimum is taken over all smooth curves γ joining z to w in D.

It is immediate from the construction of dD that it satisfies the requirementsfor a distance on D, namely(a) dD(z, w) ≥ 0 with equality if and only if z = w;(b) dD(z, w) = dD(w, z);(c) for all u, v, w in D, dD(u,w) ≤ dD(u, v) + dD(v, w).

The hyperbolic area of a Borel measurable subset of D is

aD(E) =

∫ ∫

E

λ2

D(z)dxdy.

We need to identify the isometries of both the hyperbolic metric and thehyperbolic distance. A holomorphic function f : D → D is an isometry of the

metric λD(z) |dz| if for all z in D,

(2.3) λD

(f(z)

)|f ′(z)| = λD(z),

and it is an isometry of the distance dD if, for all z and w in D,

(2.4) dD

(f(z), f(w)

)= dD(z, w).

In fact, the two classes of isometries coincide, and each isometry is a Mobiustransformation of D onto itself.


Theorem 2.1. For any holomorphic map f : D → D the following are equivalent:

(a) f is a conformal automorphism of D;

(b) f is an isometry of the metric λD;

(c) f is an isometry of the distance dD.

Proof. First, (a) implies (b). Indeed, if (a) holds, then f is of the form (2.1),and a calculation shows that

|f ′(z)|1 − |f(z)|2 =

1

1 − |z|2 ,

so (b) holds. Next, (b) implies (a). Suppose that (b) holds; that is, f is anisometry of the hyperbolic metric. Then for any conformal automorphism g ofD, h = g f is again an isometry of the hyperbolic metric. If we choose g so thath(0) = g(f(0)) = 0, then

2|h′(0)| = λD(h(0)|h′(0)| = λD(0) = 2.

Thus, h is a holomorphic self-map of D that fixes the origin and |h′(0)| = 1, soSchwarz’s Lemma implies h ∈ A(D, 0). Then f = g

−1 h is in A(D). We havenow shown that (a) and (b) are equivalent.

Second, we prove (a) and (c) are equivalent. If f ∈ A(D), then f is an isometryof the metric λD. Hence, for any smooth curve γ in D,

ℓD

(f γ

)=

∫

fγλD(w) |dw| =

∫

γ

λD

(f(z)

)|f ′(z)| |dz| = ℓD(γ).

This implies that for all z, w ∈ D, dD(f(z), f(w)) ≤ dD(z, w). Because f ∈ A(D),the same argument applies to f

−1, and hence we may conclude that f is a dD–isometry. Finally, we show that (c) implies (a). Take any f : D → D that isholomorphic and a dD–isometry. Choose any g of the form (2.1) that maps f(0)to 0 and put h = g f . Then h is holomorphic, a dD–isometry, and h(0) = 0.Thus dD

(0, h(z)

)= dD

(h(0), h(z)

)= dD(0, z). This implies that |h(z)| = |z| and

hence, that h(z) = eiθz for some θ ∈ R. Thus h ∈ A(D, 0) and, as f = g

−1 h,f is also in A(D).

In summary, relative to the hyperbolic metric and the hyperbolic distance,the group A(D) of conformal automorphism of the unit disk becomes a group ofisometries.

Theorem 2.2. The hyperbolic distance dD(z, w) in D is given by

(2.5) dD(z, w) = log1 + pD(z, w)

1 − pD(z, w)= 2 tanh−1

pD(z, w),

where the pseudo-hyperbolic distance pD(z, w) is given by

(2.6) pD(z, w) =

∣∣∣∣z − w

1 − zw

∣∣∣∣ .


Proof. First, we prove that if −1 < x < y < 1 then

(2.7) dD(x, y) = log

(1 + y−x

1−xy

1 − y−x

1−xy

).

Consider a smooth curve γ joining x to y in D, and write γ(t) = u(t) + iv(t),where 0 ≤ t ≤ 1. Then

ℓD(γ) =

∫1

0

2 |γ′(t)| dt

1 − |γ(t)|2 ≥∫

1

0

2 u′(t) dt

1 − u(t)2

because |γ(t)|2 ≥ |u(t)|2 = u(t)2 and |γ′(t)| ≥ |u′(t)| ≥ u′(t). The second integral

can be evaluated directly and gives

ℓD(γ) ≥ log

(1 + y

1 − y

1 − x

1 + x

)= log

(1 + y−x

1−xy

1 − y−x

1−xy

).

Because equality holds here when γ(t) = x + t(y − x), 0 ≤ t ≤ 1, we see that(2.7) holds, so (2.5) is valid for −1 < x < y < 1.

Now we have to extend (2.5) to any pair of points z and w in D. Theorem2.1 shows that each Euclidean rotation about the origin is a hyperbolic isometryand this implies that, for all z, dD(0, z) = dD(0, |z|). Now take any z and w inD, and let f(z) = (z −w)/(1− zw). Then f is a conformal automorphism of D,and so is a hyperbolic isometry. Thus

dD(z, w) = dD(w, z)

= dD

(f(w), f(z)

)

= dD

(0, f(z)

)

= dD

(0, |f(z)|

)

= dD

(0, pD(z, w)

),

which, from (2.7) with x = 0 and y = pD(z, w), gives (2.5).

Note that (2.5) produces

dD(0, z) = log1 + |z|1 − |z| , dD(0, z) = 2 tanh−1 |z|.

Also,

limz→w

dD(z, w)

|z − w| = λD(w) = 2 limz→w

pD(z, w)

|z − w| .

A careful examination of the proof of (2.5) shows that if γ is a smooth curvethat joins x to y, where −1 < x < y < 1, then ℓD(γ) = dD(0, x) if and only if γ

is the simple arc from x to y along the real axis. As hyperbolic isometries mapcircles into circles, map the unit circle onto itself, and preserve orthogonality, wecan now make the following definition.


Definition 2.3. Suppose that z and w are in D. Then the (hyperbolic) geodesic

through z and w is C ∩ D, where C is the unique Euclidean circle (or straightline) that passes through z and w and is orthogonal to the unit circle ∂D. If γ isany smooth curve joining z to w in D, then the hyperbolic length of γ is dD(z, w)if and only if γ is the simple arc of C in D that joins z and w.

The unit disk D together with the hyperbolic metric is called the Poincare

model of the hyperbolic plane. The “lines” in the hyperbolic plane are the hyper-bolic geodesics and the angle between two intersecting lines is the Euclidean anglebetween the Euclidean tangent lines at the point of intersection. The hyperbolicplane satisfies all of the axioms for Euclidean geometry with the exception of theParallel Postulate. It is easy to see that if γ is a hyperbolic geodesic in D anda ∈ D is a point not on γ, then there are infinitely many geodesics through a

that do not intersect γ and so are parallel to γ.

We shall now show that the hyperbolic distance dD is additive along geodesics.By contrast, the pseudo-hyperbolic distance pD is never additive along geodesics.

Theorem 2.4. If u, v and w are three distinct points in D that lie, in this order,

along a geodesic, then dD(u,w) = dD(u, v) + dD(v, w). For any three distinct

points u, v and w in D, pD(u,w) < pD(u, v) + pD(v, w).

Proof. Suppose that u, v and w lie in this order, along a geodesic. Then thereis an isometry f that maps this geodesic to the real diameter (−1, 1) of D, withf(v) = 0. Let x = f(u) and y = f(w), so that −1 < x < 0 < y < 1. It issufficient to show that dD(x, 0) + dD(0, y) = dD(x, y); this is a direct consequenceof (2.7).

It is easy to verify that pD a distance function on D, except possibly for theverification of the triangle inequality. This holds because, for any distinct u, v

and w,

pD(u,w) = tanh1

2dD(u,w)

≤ tanh1

2[dD(u, v) + dD(v, w)]

=tanh 1

2dD(u, v) + tanh 1

2dD(v, w)

1 + tanh 1

2dD(u, v) tanh 1

2dD(v, w)

< tanh1

2dD(u, v) + tanh

1

2dD(v, w)

= pD(u, v) + pD(v, w).

This also shows that there is always a strict inequality in the triangle inequalityfor pD for any three distinct points.

The following example illustrates how the hyperbolic distance compares withthe Euclidean distance in D.


Example 2.5. The Poincare model of the hyperbolic plane does not accuratelyreflect all of the properties of the hyperbolic plane. For example, the hyperbolicplane is homogeneous; this means that for any pair of points a and b in D thereis an isometry f with f(a) = b. Intuitively this means that the hyperbolic planelooks the same at each point just as the Euclidean plane does. However, withour Euclidean eyes, the origin seems to occupy a special place in the hyperbolicplane. In fact, in the hyperbolic plane the origin is no more special than anypoint a 6= 0.

Here is another way in which the Poincare model deceives our Euclidean eyes.Let x0, x1, x2, . . . be the sequence 0, 1

2,

3

4,

7

8, . . ., so that xn = (2n − 1)/2n, and

xn+1 is halfway between xn and 1 in the Euclidean sense. A computation using(2.5) shows that dD(0, xn) = log(2n+1 − 1). We conclude that dD(xn, xn+1) →log 2 as n → ∞; thus the points xn are, for large n, essentially equally spacedin the hyperbolic sense along the real diameter of D. Moreover, in any figurerepresenting the Poincare model the points xn, for n ≥ 30, are indistinguishablefrom the point 1 which does not lie in the hyperbolic plane. In brief, although thehyperbolic plane contains arbitrarily large hyperbolic disks about the origin, ourEuclidean eyes can only see hyperbolic disks about the origin with a moderatesized hyperbolic radius.

Let us comment now on the various formulae that are available for dD(z, w).It is often tempting to use the pseudohyperbolic distance pD rather than thehyperbolic distance dD (and many authors do) because the expression for pD isalgebraic whereas the expression for dD is not. However, this temptation shouldbe resisted. The distance pD is not additive along geodesics, and it does notarise from a Riemannian metric. Usually, the solution is to use the followingfunctions of dD, for it is these that tend to arise naturally and more frequentlyin hyperbolic trigonometry:

(2.8) sinh21

2dD(z, w) =

|z − w|2(1 − |z|)2(1 − |w|2) =

1

4|z − w|2λD(z)λD(w),

and

cosh21

2dD(z, w) =

|1 − zw|2(1 − |z|2)(1 − |w|2) =

1

4|1 − zw|2λD(z)λD(w).

These can be proved directly from (2.5), and together they give the familiarformula

tanh1

2dD(z, w) =

∣∣∣∣z − w

1 − zw

∣∣∣∣ = pD(z, w).

We investigate the topology defined on the unit disk by the hyperbolic dis-tance. For this we study hyperbolic disks since they determine the topology. Thehyperbolic circle Cr given by z ∈ D : dD(0, z) = r is a Euclidean circle withEuclidean center 0 and Euclidean radius tanh 1

2r. Now let C be any hyperbolic

circle, say of hyperbolic radius r and hyperbolic center w. Then there is a hyper-bolic isometry f with f(w) = 0, so that f(C) = Cr. As Cr is a Euclidean circle,so is f

−1(Cr), which is C. Conversely, suppose that C is a Euclidean circle in D.


Then there is a hyperbolic isometry f such that f(C) is a Euclidean circle withcenter 0, so that f(C) = Cr for some r. Thus, as f is a hyperbolic isometry,f−1(Cr) = C, is also a hyperbolic circle. This shows that the set of hyperbolic

circles coincides with the set of Euclidean circles in D. As the same is obviouslytrue for open disks (providing that the closed disks lie in D), we see that the

topology induced by the hyperbolic distance on D coincides with the Euclidean

topology on the unit disk.

Theorem 2.6. The topology induced by dD on D coincides with the Euclidean

topology. The space D with the distance dD is a complete metric space.

Proof. We have already proved the first statement. Suppose, then, that (zn)is a Cauchy sequence with respect to the distance dD. Then (zn) is a boundedsequence with respect to dD and, as we have seen above, this means that the(zn) lie in a compact disk K that is contained in D. As λD ≥ 2 on D, wesee immediately from (2.8) that (zn) is a Cauchy sequence with respect to theEuclidean metric, so that zn → z

∗, say, where z∗ ∈ K ⊂ D. It is now clear that

dD(zn, z∗) → 0 so that D with the distance dD is complete.

The Euclidean metric on D arises from the fact that D is embedded in thelarger space C and is not complete on D. By contrast, an important property ofthe distance dD is that dD(0, |z|) → +∞ as |z| → 1; informally, the boundary ∂D

of D is ‘infinitely far away’ from each point in D. This is a consequence of thefact that D equipped with the hyperbolic distance dD is a complete metric spaceand is another reason why dD should be preferred to the Euclidean metric on D.

Exercises.

1. Verify that (2.1) and (2.2) determine the same subgroup of Mobius trans-formations.

2. Suppose equality holds in the triangle inequality for the hyperbolic distance;that is, suppose u, v, w in D and dD(u,w) = dD(u, v) + dD(v, w). Provethat u, v and w lie on a hyperbolic geodesic in this order.

3. Verify that the hyperbolic disk DD(a, r) is the Euclidean disk with centerc and radius R, where

c =a

(1 − tanh2(r/2)

)

1 − |a|2 tanh2(r/2)and R =

(1 − |a|2) tanh(r/2)

1 − |a|2 tanh2(r/2).

4. (a) Prove that the hyperbolic area of a hyperbolic disk of radius r is4π sinh2(r/2).(b) Show that the hyperbolic length of a hyperbolic circle with radius r is2π sinh r.

3. The Schwarz-Pick Lemma

We begin with a statement of the classical Schwarz Lemma.


Theorem 3.1 (Schwarz’s Lemma). Suppose that f : D → D is holomorphic and

that f(0) = 0. Then either

(a) |f(z)| < |z| for every non-zero z in D, and |f ′(0)| < 1, or

(b) for some real constant θ, f(z) = eiθz and |f ′(0)| = 1.

The Schwarz Lemma is proved by applying the Maximum Modulus Theoremto the holomorphic function f(z)/z on the unit disk D. It says that if a holo-morphic function f : D → D fixes 0 then either (a) f(z) is closer to 0 than z is,or (b) f is a rotation of the plane about 0. Although both of these assertions aretrue in the context of Euclidean geometry, they are only invariant under confor-mal maps when they are interpreted in terms of hyperbolic geometry. Moreover,as Pick observed in 1915, in this case the requirement that f has a fixed pointin D is redundant. We can now state Pick’s invariant formulation of Schwarz’sLemma [33].

Theorem 3.2 (The Schwarz-Pick Lemma). Suppose that f : D → D is holomor-

phic. Then either

(a) f is a hyperbolic contraction; that is, for all z and w in D,

(3.1) dD

(f(z), f(w)

)< dD(z, w), λD

(f(z)

)|f ′(z)| < λD(z),

or

(b) f is a hyperbolic isometry; that is, f ∈ A(D) and for all z and w in D,

(3.2) dD

(f(z), f(w)

)= dD(z, w), λD

(f(z)

)|f ′(z)| = λD(z)

Proof. By Theorem 2.1, f is an isometry if and only if one, and hence both,of the conditions in (3.2) hold. Suppose now that f : D → D is holomorphicbut not an isometry. Select any two points z1 and z2 in D. Here is the intuitiveidea behind the proof. Because the hyperbolic plane is homogeneous, we mayassume without loss of generality that both z1 and f(z1) are at the origin. Inthis special situation (3.1) follows directly from part (b) of Theorem 3.1. Nowwe write out a formal argument. Let g and h be conformal automorphisms (andhence isometries) of D such that g(z1) = 0 and h

(f(z1)

)= 0. Let F = hfg

−1;then F is a holomorphic self-map of D that fixes 0. As g and h are isometries,F is not an isometry or else f would be too. Therefore, by Schwarz’s Lemma,for all z, dD

(0, F (z)

)< dD(0, z) and |F ′(0)| < 1. Thus, as Fg = hf and g, h are

hyperbolic isometries,

dD

(f(z1), f(z2)

)= dD

(hf(z1), hf(z2)

)

= dD

(Fg(z1), Fg(z2)

)

= dD

(0, Fg(z2)

)

< dD

(0, g(z2)

)

= dD

(g(z1), g(z2)

)

= dD(z1, z2).


This is the first inequality in (3.1). To obtain the second inequality, we applythe Chain Rule to each side of Fg = hf and obtain

|F ′(0)| =|f ′(z1)|(1 − |z1|2)

1 − |f(z1)|2< 1.

This gives the second inequality in (3.1) at an arbitrary point z1.

Often the Schwarz-Pick Lemma is stated in the following form: Every holo-morphic self-map of the unit disk is a contraction relative to the hyperbolicmetric. That is, if f is a holomorphic self-map of D, then

(3.3) dD

(f(z), f(w)

)≤ dD(z, w), λD

(f(z)

)|f ′(z)| ≤ λD(z).

If equality holds in either inequality, then f is a conformal automorphism of D.One should note that the two inequalities in (3.3) are equivalent. If the firstinequality holds, then

λD

(f(z)

)|f ′(z)| = lim

w→z

dD(f(z), f(w))

|f(z) − f(w)||f(z) − f(w)|

|z − w| ≤ limw→z

dD(z, w)

|z − w| = λD(z).

On the other hand, if the second inequality holds, then integration over any pathγ in D gives ℓD(f γ) ≤ ℓD(γ). This implies the first inequality in (3.3).

Hyperbolic geometry had been used in complex analysis by Poincare in hisproof of the Uniformization Theorem for Riemann surfaces. The work of Pick isa milestone in geometric function theory, it shows that the hyperbolic metric, notthe Euclidean metric, is the natural metric for much of the subject. The definitionof the hyperbolic metric might seem arbitrary. In fact, up to multiplication by apositive scalar it is the only metric on the unit disk that makes every holomorphicself-map a contraction, or every conformal automorphism an isometry.

Theorem 3.3. For a metric ρ(z)|dz| on the unit disk the following are equivalent:

(a) For any holomorphic self-map of D and all z ∈ D, ρ(f(z))|f ′(z)| ≤ ρ(z);(b) For any f ∈ A(D) and all z ∈ D, ρ(f(z))|f ′(z)| = ρ(z);(c) ρ(z) = cλD for some c > 0.

Proof. (a)⇒(b) Suppose f ∈ A(D). Then the inequality in (a) holds for f . Theinequality in (a) also holds for f

−1; this gives ρ(z) ≤ ρ(f(z))|f ′(z)|. Hence, everyconformal automorphism of D is an isometry relative to ρ(z)|dz|.

(b)⇒(c) Define c > 0 by ρ(0) = cλD(0). Now, consider any a ∈ D. Let f bea conformal automorphism of D with f(0) = a. Then because f is an isometryrelative to both ρ(z)|dz| and the hyperbolic metric,

ρ(a)|f ′(0)| = ρ(0)

= cλD(0)

= cλD(a)|f ′(0)|.Hence, ρ(a) = cλD(a) for all a ∈ D.

(c)⇒(a) This is an immediate consequence of the Schwarz-Pick Lemma.


Exercises.

1. Suppose f is a holomorphic self-map of the unit disk. Prove |f ′(0)| ≤ 1.Determine a necessary and sufficient condition for equality.

2. If a holomorphic self-map of the unit disk fixes two points, prove it is theidentity.

3. Let a and b be distinct points in D.(a) Show that there exists a conformal automorphism f of D that inter-changes a and b; that is, f(a) = b and f(b) = a.(b) Suppose a holomorphic self-map f of D interchanges a and b; that is,f(a) = b and f(b) = a. Prove f is a conformal automorphism with order2, or f f is the identity.

4. An extension of the Schwarz-Pick Lemma

Recently, the authors [8] established a multi-point version of the Schwarz-PickLemma that unified a number of known variations of the Schwarz and Schwarz-Pick Lemmas and also has many new consequences. A selection of results from[8] are presented in this and the next section; for more results of this type, consultthe original paper.

We begin with a brief discussion of Blaschke products. A function F : D → D

is a (finite) Blaschke product if it is holomorphic in D, continuous in D (the closedunit disk), and |F (z)| = 1 when |z| = 1. If F is a Blaschke product then so arethe compositions g(F (z)) and F (g(z)) for any conformal automorphism g of D.In addition, it is clear that any finite product of conformal automorphisms of D isa Blaschke product. We shall now show that the converse is true. Suppose thatF is a Blaschke product. If F has no zeros in D then, by the Minimum ModulusTheorem, F is a constant, which must be of modulus one. Now suppose that F

does have a zero in D. Then it can only have a finite number of zeros in D, saya1, . . . , ak (which need not be distinct), and

F (z)

/ k∏

m=1

(z − am

1 − amz

)

is a Blaschke product with no zeros in D. This shows that F is a Blaschke

product if and only if it is a finite product of automorphisms of D. We say thatF is of degree k if this product has exactly k non-trivial factors.

We now discuss the complex pseudo-hyperbolic distance in D, and the hyper-bolic equivalent of the usual Euclidean difference quotient of a function.

Definition 4.1. The complex pseudo-hyperbolic distance [z, w] between z and w

in D is given by

[z, w] =z − w

1 − wz.

We recall that the pseudo-hyperbolic distance is |[z, w]|; see (2.6).


The complex pseudo-hyperbolic distance is an analog for the hyperbolic planeD of the real directed distance x − y from y to x, for points on the real line R.

Definition 4.2. Suppose that f : D → D is holomorphic, and that z, w ∈ D

with z 6= w. The hyperbolic difference quotient f∗(z, w) is given by

f∗(z, w) =

[f(z), f(w)]

[z, w].

If we combine (2.6) with the Schwarz-Pick Lemma we see that

pD

(f(z1), f(z2)

)≤ pD(z1, z2),

and that equality holds for one pair z1 and z2 of distinct points if and only if f isa conformal automorphism of D (in which case, equality holds for all z1 and z2).It follows that if f : D → D is holomorphic, then either f is a hyperbolic isometryand |f ∗(z, w)| = 1 for all z and w, or f is not an isometry and |f ∗(z, w)| < 1 forall z and w.

We shall now discuss the hyperbolic difference quotient f∗(z, w). This is a

function of two variables but, unless we state explicitly to the contrary, we shall

regard it as a holomorphic function of the single variable z. Note that f∗(z, w)

is not holomorphic as a function of the second variable w. The basic propertiesof f

∗(z, w) are given in our next result.

Theorem 4.3. Suppose that f : D → D is holomorphic, and that w ∈ D.

(a) The function z 7→ f∗(z, w) is holomorphic in D.

(b) If f is not a conformal automorphism of D, then z 7→ f∗(z, w) is a holo-

morphic self-map D.

(c) The map z 7→ f∗(z, w) is a conformal automorphism of D if and only if f

is a Blaschke product of degree two.

Proof. Part (a) is obvious as w is a removable singularity of the function

f∗(z, w) =

(f(z) − f(w)

1 − f(w)f(z)

)(z − w

1 − wz

)−1

.

Now suppose that f is not a conformal automorphism of D. Then, as we haveseen above, |f ∗(z, w)| < 1 and this proves (b).

To prove (c) we note first that there are conformal automorphisms g and h

(that depend on w) of D such that f∗(z, w) = g(f(z))/h(z) or, equivalently,

f(z) = g−1(f∗(z, w)h(z)

). Clearly, if f

∗(z, w) is an automorphism then f isa Blaschke product of degree two. Conversely, suppose that f is a Blaschkeproduct of degree two. Then g

(f(z)

)is also a Blaschke product, say B, of

degree two and f∗(z, w) = B(z)/h(z). As f

∗(z, w) is holomorphic in z, we seethat B(z) = h(z)h1(z) for some automorphism h1. Thus f

∗(z, w) = h1(z) asrequired.

We shall now derive a three-point version of the Schwarz-Pick Lemma. Becauseit involves three points rather than two points as in the Schwarz-Pick Lemma, the


following theorem has extra flexibility and it includes all variations and extensionsof the Schwarz-Pick Lemma that are known to the authors. We stress, though,that this theorem contains much more than simply the union of all such knownresults. Although several Euclidean variations of the Schwarz-Pick Lemma areknown, in our view much greater clarity is obtained by a strict adherence to hy-perbolic geometry. This and other stronger versions of the Schwarz-Pick Lemmaappear in [8].

Theorem 4.4 (Three-point Schwarz-Pick Lemma). Suppose that f is holomor-

phic self-map of D, but not an automorphism of D. Then, for any z, w and v in

D,

(4.1) dD

(f∗(z, v), f ∗(w, v)

)≤ dD(z, w).

Further, equality holds in (4.1) for some choice of z, w and v if and only if f is

a Blaschke product of degree two.

Proof. As f is holomorphic in D, but not an automorphism, Theorem 4.3(b)shows that the left-hand side of (4.1) is defined. The inequality (4.1) now followsby applying the Schwarz-Pick Lemma to the holomorphic self-map z 7→ f

∗(z, v)of D. The Schwarz-Pick Lemma also implies that equality holds in (4.1) if andonly if f

∗(z, w) is a conformal automorphism of D and, by Theorem 4.3(c), thisis so if and only if f is a Blaschke product of degree two.

Theorem 4.4 is a genuine improvement of the Schwarz-Pick Lemma. Suppose,for example that f : D → D is holomorphic, but not an automorphism, and thatf(0) = 0. Then the Schwarz-Pick Lemma tells us only that f(z)/z lies in thehyperbolic plane D, and that |f ′(0)| < 1. However, it we put w = 0 in (4.1),and then let v → 0, we obtain the stronger conclusion that f(z)/z lies in thehyperbolic disk with center f

′(0) and hyperbolic radius dD(0, z).

Exercises.

1. If f(z) is a Blaschke product of degree k, prove that f∗(z, w) is a Blaschke

product of degree k − 1.

2. Verify the following Chain Rule for the ∗-operator: For all z and w in D,and all holomorphic maps f and g of D into itself,

(f g)∗(z, w) = f∗(

g(z), g(w))g∗(z, w).

5. Hyperbolic derivatives

Since the hyperbolic metric is the natural metric to study holomorphic self-maps of the unit disk, one should also use derivatives that are compatible withthis metric. We begin with the definition of a hyperbolic derivative; just asthe Euclidean difference quotient leads to the usual Euclidean derivative, thehyperbolic difference quotient results in the hyperbolic derivative.


Definition 5.1. Suppose that f : D → D is holomorphic, but not an isometryof D. The hyperbolic derivative f

h(w) of f at w in D is

fh(w) = lim

z→w

[f(z), f(w)]

[z, w]=

(1 − |w|2)f ′(w)

1 − |f(w)|2 .

The hyperbolic distortion of f at w is

|fh(w)| = limz→w

dD(f(z), f(w))

dD(z, w).

By Theorem 4.3, |fh(z)| ≤ 1, and equality holds for some z if and only ifequality holds for all z, and then f is a conformal automorphism of D. Theorem4.4 leads to the following upper bound on the magnitude of the hyperbolic dif-ference quotient in terms of dD(z, w) and the derivative at any point v betweenz and w.

Theorem 5.2. Suppose that f : D → D is holomorphic. Then, for all z and w

in D, and for all v on the closed geodesic arc joining z and w,

(5.1) dD

(0, f ∗(z, w)

)≤ dD

(0, fh(v)

)+ dD(z, w).

Proof. First, it is clear that for any z and w, |f ∗(z, w)| = |f ∗(w, z)|. Thus

dD

(0, f ∗(z, w)

)= dD

(0, f ∗(w, z)

).

Next, Theorem 4.4 (applied twice) gives

dD

(0, f ∗(z, w)

)≤ dD

(0, f ∗(v, w)

)+ dD

(f∗(v, w), f ∗(z, w)

)

≤ dD

(0, f ∗(v, w)

)+ dD(z, v)

= dD

(0, f ∗(w, v)

)+ dD(z, v)

≤ dD

(0, f ∗(u, v)

)+ dD(w, u) + dD(z, v).

We now let u → v, where v lies on the geodesic between z and w, and asdD(z, v) + dD(v, w) = dD(z, w), we obtain (5.1).

Our next task is to transform (5.1) into a more transparent inequality aboutf . This is the next result which we may interpret as a Hyperbolic Mean Value

Inequality, a result from [7].

Theorem 5.3 (Hyperbolic Mean Value Inequality). Suppose that f : D → D is

holomorphic. Then, for all z and w in D, and for all v on the closed geodesic

arc joining z and w,

(5.2) dD

(f(z), f(w)

)≤ log

(cosh dD(z, w) + |fh(v)| sinh dD(z, w)

).

This inequality is sharper than the Schwarz-Pick inequality for if we use|fh(v)| ≤ 1 and the identity cosh t + sinh t = e

t, we recapture the Schwarz-Pick inequality. It is known that equality holds in (5.2) if and only if f is aBlaschke product of degree two and has a unique critical point c, such that ei-ther c, z = v, w, or c, w = v, z, lie in this order along a geodesic. We refer


the reader to [8] for a proof of this, and for the fact that a Blaschke product ofdegree two has exactly one critical point in D.

Proof. First, we note that, for all u and v,

tanh1

2dD(0, u) = |u|,

tanh1

2dD(u, v) = pD(u, v) = tanh

1

2dD(0, [u, v]).

Next, using the definition of f∗(z, w), the inequality in Theorem 5.2, and the

addition formula for tanh(s + t), we have

tanh1

2dD(f(z), f(w)) = pD(f(z), f(w))

= pD(z, w)|f ∗(z, w)|

= pD(z, w) tanh1

2dD

(0, f ∗(z, w)

)

≤ pD(z, w) tanh[12dD

(0, fh(v)

)+

1

2dD(z, w)

]

= pD(z, w)

(pD(z, w) + |fh(v)|1 + pD(z, w)|fh(v)|

).

Now the increasing function x 7→ tanh(1

2x) has inverse x 7→ log(1 + x)/(1 − x),

so we conclude that, with p = pD(z, w) and d = |fh(v)|,

dD(f(z), f(w)) ≤ log

(1 + pd + p(d + p)

1 + pd − p(d + p)

)= log

(1 + p

2

1 − p2+ d

2p

1 − p2

),

which is (5.2).

Next, we provide a Schwarz-Pick type of inequality for hyperbolic derivatives;recall that the hyperbolic derivative is not holomorphic. This result is basedon the observation that if f : D → D is holomorphic, but not a conformalautomorphism of D, then f

h(z) and fh(w) lie in D so that we can measure the

hyperbolic distance between these two hyperbolic derivatives.

Theorem 5.4. Suppose that f : D → D is holomorphic but not a conformal

automorphism of D. Then, for all z and w in D,

(5.3) dD

(f

h(z), fh(w))≤ 2dD(z, w) + dD

(f∗(z, w), f ∗(w, z)

).

Proof. Theorem 4.4 implies that for all z, w and v,

dD

(f∗(z, w), f ∗(v, w)

)≤ dD(z, v).

We let v → w and obtain

dD

(f∗(z, w), fh(w)

)≤ dD(z, w)

and (by interchanging z and w),

dD

(f∗(w, z), fh(z)

)≤ dD(z, w).

These last two inequalities and the triangle inequality yields (5.3).


It is easy to see that if f(0) = 0 then f∗(z, 0) = f

∗(0, z) = f(z)/z. Thus wehave the following corollary originally established in [6].

Corollary 5.5. Suppose that f : D → D is holomorphic but not a conformal

automorphism of D, and that f(0) = 0. Then, for all z,

(5.4) dD

(f

h(0), fh(z))≤ 2dD(0, z),

and the constant 2 is best possible.

Example 5.6. The preceding corollary is sharp for f(z) = z2. Note that

fh(z) = 2z/(1 + |z|2) and dD(fh(z), fh(w)) = 2dD(z, w) whenever z, w lie on

the same hyperbolic geodesic through the origin. Thus, z 7→ fh(z) doubles all

hyperbolic distances along geodesics through the origin; this doubling is notvalid in general because in hyperbolic geometry there are no similarities exceptisometries. Moreover, it is possible to verify that there is no finite K such thatdD(fh(z), fh(w)) ≤ KdD(z, w) for all z, w ∈ D, so z 7→ f

h(z) does not even sat-isfy a hyperbolic Lipschitz condition, so (5.4) is no longer valid when the originis replaced by an arbitrary point of the unit disk.

In spite of Example 5.6 a full-fledged result of Schwarz-Pick type is valid forthe hyperbolic distortion.

Corollary 5.7 (Schwarz-Pick Lemma for Hyperbolic Distortion). Suppose that

f : D → D is holomorphic but not a conformal automorphism of D. Then for all

z, w ∈ D, dD

(|fh(z)|, |fh(w)|

)≤ 2dD(z, w).

Proof. Note that, from the proof of Theorem 5.4,

dD

(|f ∗(z, w)|, |fh(w)|

)≤ dD

(f∗(z, w), fh(w)

)≤ dD(z, w),

and, similarly, dD

(|f ∗(w, z)|, |fh(z)|

)≤ dD(z, w). As |f ∗(w, z)| = |f ∗(z, w)|, the

desired inequality follows.

Exercises.

1. Verify the claims in Example 5.6.2. Suppose that f : D → D is holomorphic but not a conformal automorphism

of D. Prove that for all conformal automorphisms S and T of D, and all z

and w in D,|(S f T )∗(z, w)| = |f ∗(T (z), T (w))|.

In particular, deduce that the hyperbolic derivative is invariant in the sensethat

|(S f T )h(z)| = |fh(T (z))|.

6. The hyperbolic metric on simply connected regions

There are several equivalent definitions of what it means for a region in thecomplex plane to be simply connected. A region Ω in C is simply connected ifand only if any one of the following (equivalent) conditions hold:


(a) the set C∞\Ω is connected;(b) if f is holomorphic and never zero in Ω, then there is a single-valued holo-

morphic choice of log f in Ω;(c) each closed curve in Ω can be continuously deformed within Ω to a point

of Ω.

A region in C∞ is simply connected if (a) or (c) holds. The regions D, C andC∞ are all simply connected; an annulus is not.

Two subregions regions of C are conformally equivalent if there is a holomor-phic bijection of one onto the other. This is an equivalence relation on the classof subregions of C, and the fundamental result about simply connected regionsis the Riemann Mapping Theorem.

Theorem 6.1 (The Riemann Mapping Theorem). A subregion of C is confor-

mally equivalent to D if and only if Ω is a simply connected proper subregion of

C. Moreover, given a ∈ Ω there is a unique conformal mapping f : Ω → D such

that f(a) = 0 and f′(a) > 0.

The Riemann Mapping Theorem enables us to transfer the hyperbolic metricfrom D to any simply connected proper subregion Ω of C.

Definition 6.2. Suppose that f is a conformal map of a simply connected planeregion Ω onto D. Then the hyperbolic metric λΩ(z)|dz| of Ω is defined by

(6.1) λΩ(z) = λD

(f(z)

)|f ′(z)|.

The hyperbolic distance dΩ is the distance function on Ω derived from the hy-perbolic metric.

We need to show λΩ is independent of the choice of the conformal map f thatis used in (6.1), for this will imply that λΩ is determined by Ω alone. Suppose,then, that f is a conformal map of Ω onto D. Then the set of all conformalmaps of Ω onto D is given by h f , where h ranges over A(D). Any conformalautomorphism h of D is a hyperbolic isometry, so that for all w in D,

λD(w) = λD

(h(w)

)|h′(w)|.

If we now let g = h f , w = f(z) and use the Chain Rule we find that

λD

(g(z)

)|g′(z)| = λD

(h(f(z)

)|h′(f(z))||f ′(z)|

= λD(f(z))|f ′(z)|so that λΩ as defined in (6.1) is independent of the choice of the conformal mapf .

Thus, Definition 6.2 converts every conformal map of a simply connectedproper subregion of C onto the unit disk into an isometry of the hyperbolicmetric. The hyperbolic distance dΩ on a simply connected proper subregion Ω ofC can be defined in two equivalent ways. First, one can pull-back the hyperbolicdistance on D to Ω by setting dΩ(z, w) = dD(f(z), f(w)) for any conformal mapf : Ω → D and verifying that this is independent of the choice of the conformal


mapping onto the unit disk. Alternatively, the hyperbolic length of a path γ inΩ is

ℓΩ(γ) =

∫

γ

λΩ(z)|dz|,

and one can define

dΩ(z, w) = inf ℓΩ(γ),

where the infimum is taken over all piecewise smooth curves γ in Ω that joinz and w. These two definitions of the hyperbolic distance are equivalent. Thehyperbolic distance dΩ on Ω is complete. Moreover, a path γ in Ω connecting z

and w is a hyperbolic geodesic in Ω if and only if f γ is a hyperbolic geodesicin D. Also, for any a ∈ Ω and r > 0, f(DΩ(a, r)) = DD(f(a), r).

In fact, the essence of Definition 6.2 is that the entire body of geometricfacts about the Poincare model D of the hyperbolic plane transfers, without anyessential change, to an arbitrary simply connected proper subregion of C withits own hyperbolic metric. If f : Ω → D is any conformal mapping, then f is anisometry relative to the hyperbolic metrics and hyperbolic distances on Ω andD. The next result is an immediate consequence of Definition 6.2 and we omitits proof; it asserts that all conformal maps of simply connected proper regionsare isometries relative to the hyperbolic metrics and hyperbolic distances of theregions.

Theorem 6.3 (Conformal Invariance). Suppose that Ω1 and Ω2 are simply con-

nected proper subregions of C, and that f is a conformal map of Ω1 onto Ω2.

Then f is a hyperbolic isometry, so that for any z in Ω1,

(6.2) λΩ2

(f(z)

)|f ′(z)| = λΩ1

(z),

and for all z, w ∈ Ω1

dΩ2(f(z), f(w)) = dΩ1

(z, w).

Note that if γ is a smooth curve in Ω1, then (6.2) implies

ℓΩ2(f γ) = ℓΩ1

(γ).

Theorem 6.3 implies that each element of A(Ω), the group of conformal auto-morphisms of Ω, is a hyperbolic isometry.

Theorem 6.4 (Schwarz-Pick Lemma for Simply Connected Regions). Suppose

that Ω1 and Ω2 are simply connected proper subregions of C, and that f is a

holomorphic map of Ω1 into Ω2. Then either

(a) f is a hyperbolic contraction; that is, for all z and w in Ω1,

dΩ2

(f(z), f(w)

)< dΩ1

(z, w), λΩ2

(f(z)

)|f ′(z)| < λΩ1

(z),

or

(b) f is a hyperbolic isometry; that is, f is a conformal map of Ω1 onto Ω2 and

for all z and w in Ω1,

dΩ2

(f(z), f(w)

)= dΩ1

(z, w), λΩ2

(f(z)

)|f ′(z)| = λΩ1

(z).


Proof. Because of Theorem 6.3 we only need verify (a) when the holomorphicmap f : Ω1 → Ω2 is not a holomorphic bijection. Choose any point z0 inΩ1, and let w0 = f(z0). Next, construct a holomorphic bijection h of D, and aholomorphic bijection g of Ω1 onto Ω2; these can be constructed so that h(0) = z0

and g(z0) = w0 = f(z0). Now let k = (gh)−1fh. Then k is a holomorphic map

of D into itself and k(0) = 0. Moreover, k is not a conformal automorphism ofD or else f would be a holomorphic bijection. Thus |k′(0)| < 1 and, using theChain Rule, this gives |f ′(z0)| < |g′(z0)|. With this,

λΩ2

(f(z0)

)|f ′(z0)| < λΩ1

(z0)

follows as (6.2) holds (with f replaced by g).

This establishes the second strict inequality in (a); the first strict inequalityfor hyperbolic distances follows by integrating the strict inequality for hyperbolicmetrics.

This version of the Schwarz-Pick Lemma can be stated in the following equiv-alent form. If f : Ω1 → Ω2 is holomorphic, then for all z and w in Ω1,

(6.3) dΩ2(f(z), f(w)) ≤ dΩ1

(z, w),

and

(6.4) λΩ2

(f(z)

)|f ′(z)| ≤ λΩ1

(z).

Further, if either equality holds in (6.3) for a pair of distinct points or at onepoint z in (6.4) , then f is a conformal bijection of Ω1 onto Ω2.

Corollary 6.5 (Schwarz’s Lemma for Simply Connected Regions). Suppose Ωis a simply connected proper subregion of Ω and a ∈ Ω. If f is a holomorphic

self-map of Ω that fixes a, then |f ′(a)| ≤ 1 and equality holds if and only if

f ∈ A(Ω, a), the group of conformal automorphisms of Ω that fix a. Moreover,

f′(a) = 1 if and only if f is the identity.

Theorem 6.4 is the fundamental reason for the existence of many distortiontheorems in complex analysis. Consider the class of holomorphic maps of Ω1 into

Ω2. Then any such map f will have to satisfy the universal constraints (6.3) and(6.4) where the metrics λΩ1

and λΩ2are uniquely determined (albeit implicitly)

by the regions Ω1 and Ω2. Thus (6.3) and (6.4) are, in some sense, the genericdistortion theorems for holomorphic maps.

This is the appropriate place to point out that neither the complex plane C

nor the extended complex plane C∞ has a metric analogous to the hyperbolicmetric in the sense that the metric is invariant under the group of conformalautomorphisms. Recall that A(C) is the set of all maps z 7→ az + b, a, b ∈ C anda 6= 0, and A(C∞) is the group M of Mobius transformations. The group A(C)acts doubly transitively on C; that is, given two pairs z1, z2 and w1, w2 of distinctpoints in C there is a conformal automorphism f of C with f(zj) = wj, j = 1, 2.Similarly, M acts triply transitively on C∞. If there were a conformal metricon either C or C∞ invariant under the full conformal automorphism group, then


the distance function induced from this metric would also be invariant under theaction of the full group of conformal automorphisms. The following result showsthat only trivial distance functions are invariant under A(C) or A(C∞).

Theorem 6.6. If d is a distance function on C or C∞ that is invariant under

the full group of conformal automorphisms, then there exists t > 0 such that

d(z, w) = 0 if z = w and d(z, w) = t otherwise.

Proof. Let d be a distance function on C that is invariant under A(C). Set t =d(0, 1). Consider any distinct z, w ∈ C. Because A(C) acts doubly transitivelyon C, there exists f ∈ A(C) with f(0) = z and f(1) = w. The invariance of d

under A(C) implies d(z, w) = d(f(0), f(1)) = d(0, 1) = t. The same argumentapplies to C∞.

The Euclidean metric |dz| on C is invariant under the proper subgroup ofA(C) given by z 7→ az + b, where |a| = 1 and b ∈ C. The spherical metric2|dz|/(1 + |z|2) on C∞ is invariant under the group of rotations of C∞, that is,Mobius maps of the form

z 7→ az − c

cz + a, a, c ∈ C, |a|2 + |c|2 = 1,

or of the equivalent form

z 7→ eiθ z − a

1 + az, θ ∈ R, a ∈ C∞.

The group of rotations of C∞ is a proper subgroup of M.

Exercises.

1. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. LetF denote the family of all holomorphic functions f defined on D such thatf(D) ⊆ Ω and f(0) = a. Set M = sup|f ′(0)| : f ∈ F. Prove M < +∞and that |f ′(0)| = M if and only if f is a conformal map of D onto Ω withf(0) = a. Show M = 2/λΩ(a).

2. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. LetG denote the family of all holomorphic functions f defined on Ω such thatf(Ω) ⊆ D. Set N = sup|f ′(a)| : f ∈ G. Prove N < +∞ and that|f ′(a)| = N if and only if f is a conformal map of Ω onto D with f(a) = 0.Show N = λΩ(a)/2.

3. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. LetH(Ω, a) denote the family of all holomorphic self-maps of Ω that fix a.Prove that f ′(a) : f ∈ H(Ω, a) equals the closed unit disk.

7. Examples of the hyperbolic metric

We give examples of simply connected regions and their hyperbolic metrics.These metrics are computed by using (6.2) in the following way: one finds anexplicit conformal map f from the region Ω1 whose metric is sought onto a region


Ω2 whose metric is known. Then (6.2) enables one to find an explicit expressionfor λΩ1

(z) for z in Ω1. We omit almost all of the computations.

The simplest instance of the Riemann Mapping Theorem is the fact thatany disk or half-plane is Mobius equivalent to the unit disk. Because hyperboliccircles (disks) in D are Euclidean circle (disks) in D, we deduce that an analogousresult holds for any disk or half-plane. Also, in any disk or half-plane hyperbolicgeodesics are arcs of circles orthogonal to the boundary; in the case of a half-planes we allow half-lines orthogonal to the edge of the half-plane.

Example 7.1 (disk). As f(z) = (z − z0)/R is a conformal map of the diskD = z : |z − z0| < R onto D, we find

λD(z)|dz| =2R |dz|

R2 − |z − z0|2.

In particular,

λD(z0) =2

R.

Example 7.2 (half-plane). Let H be the upper half-plane x+iy : y > 0. Theng(H) = D, where g(z) = (z − i)/(z + i), so H = x + iy : y > 0 has hyperbolicmetric

λH(z)|dz| =|dz|y

=|dz|Im z

.

Similarly, the hyperbolic metric of the right half-plane K = x + iy : x > 0 is|dz|/x. More generally, if H is any open half-plane, then

λH(z)|dz| =|dz|

d(z, ∂H),

where d(z, ∂H) denotes the Euclidean distance from z to ∂H.

Theorem 7.3. If f : D → K is holomorphic and f(0) = 1, then

(7.1)1 − |z|1 + |z| ≤ Re f(z) ≤ 1 + |z|

1 − |z|and

(7.2) |Im f(z)| ≤ 2|z|1 − |z|2

Proof. This is an immediate consequence of the Schwarz-Pick Lemma after con-verting the conclusion into weaker Euclidean terms. Fix z ∈ D and set

r = dD(0, z) = 2 tanh−1 |z| = log1 + |z|1 − |z| .

The Schwarz-Pick Lemma implies that f(z) lies in the closed hyperbolic diskDK(1, r). The closed hyperbolic disk DK(1, r) has Euclidean center cosh r, Eu-clidean radius sinh r and the bounding circle meets the real axis at e

−r and er;


see the exercises. Therefore, f(z) lies in the closed Euclidean square z = x+iy :e−r ≤ x ≤ e

r, |y| ≤ sinh r. Since

e−r =

1 − |z|1 + |z| and e

r =1 + |z|1 − |z| ,

(7.1) is established. Finally,

sinh r =2|z|

1 − |z|2demonstrates (7.2).

Theorem 7.4. Suppose that H is any disk or half-plane. Then for all z and w

in H,

sinh21

2dH(z, w) = 1

4|z − w|2λH(z)λH(w).

Proof. It is easy to verify that for any Mobius map g we have

(7.3)(g(z) − g(w)

)2= (z − w)2

g′(z) g

′(w).

Now take any Mobius map g that maps H onto D, and recall that g is an isometryfrom H to D if both are given their hyperbolic metrics. Then, using (2.8) and(7.3)

1

4|z − w|2λH

(z

)λH(w

)= 1

4|z − w|2λD

(g(z)

)λD

(g(w)

)|g′(z)| |g′(w)|

= 1

4|g(z) − g(w)|2λD(g(z))λD(g(w))

= sinh21

2dD(g(z), g(w))

= sinh21

2dH(z, w

)

There is another, less well known, version of the Schwarz-Pick Theorem avail-able which is an immediate consequence of Theorem 7.4, and which we state ina form that is valid for all disks and half-planes.

Theorem 7.5 (Modified Schwarz-Pick Lemma for Disks and Half-Planes). Sup-

pose that Hj is any disk or half-plane, j = 1, 2, and that f : H1 → H2 is

holomorphic. Then, for all z and w in H1,

|f(z) − f(w)|2|z − w|2 ≤ λH1

(z)λH1(w)

λH2

(f(z)

)λH2

(f(w)

) .

Proof. By Theorem 7.4 and the Schwarz-Pick Lemma

1

4|f(z) − f(w)|2λH2

(f(z))λH2(f(w)) = sinh2

1

2dH2

(f(z), f(w))

≤ sinh21

2dH1

(z, w)

= 1

4|z − w)|2λH1

(z

)λH1

(w

).


Observe that if w → z in Theorem 7.5, then we obtain (6.4) in the special caseof disks and half-planes. We give an application of Theorem 7.5 to holomorphicfunctions.

Example 7.6. Suppose that f is holomorphic in the open unit disk and that f

has positive real part. Then f maps D into K, and we have

|f(z) − f(w)|2|z − w|2 ≤ 4 Re [f(z)] Re [f(w)]

(1 − |z|2)(1 − |w|2) .

This implies, for example, that if we also have f(0) = 1 then |f ′(0)| ≤ 2.

Example 7.7 (slit plane). Since f(z) =√

z maps P = C\x ∈ R : x ≤ 0 ontoK = x + iy : x > 0, the hyperbolic metric on P is

λP (z) |dz| =|dz|

2|√z|Re [√

z].

This gives

λP (z) =1

2r cos(θ/2)≥ 1

2|z| ,

where z = reiθ.

Example 7.8 (sector). Let S(α) = z : 0 < arg(z) < απ, where 0 < α ≤ 2.Here, f(z) = z

1/α = exp(α−1 log z

)is a conformal map of S(α) onto H, so S(α)

has hyperbolic metric

λS(α)(z) |dz| =|z|1/α

α|z| Im[z1/α]|dz|.

Note that this formula for the hyperbolic metric agrees (as it must) with theformula for λH in Example 7.2 (which is the case α = 1). The special case α = 2is the preceding example.

Example 7.9 (doubly infinite strip). S = x + iy : |y| < π/2 has hyperbolicmetric

λS(z) |dz| =|dz|cos y

.

In this case we use the fact that ez maps S conformally onto K = x + iy : x >

0. Notice that λS(z) ≥ 1 with equality if and only if z lies on the real axis.In particular, the hyperbolic distance between points on R is the same as theEuclidean distance between the points.

Theorem 7.10. Let S = z : |Im(z)| < π/2. Then for any a ∈ R and any

holomorphic self-map f of S, |f ′(a)| ≤ 1. Moreover, f′(a) = 1 if and only if

f(z) = z + c for some c ∈ R and f′(a) = −1 if and only if f(z) = −z + c for

some c ∈ R. In particular, for any interval [a, b] in R, the Euclidean length of

the image f([a, b]) is at most b − a.


Proof. From Example 7.9 for z ∈ S

λS(z) =1

cos y≥ 1

and equality holds if and only if Im(z) = 0. This observation together with theSchwarz-Pick Lemma gives

|f ′(a)| ≤ λS(f(a))|f ′(a)| ≤ λS(a) = 1

and equality implies f(a) ∈ R. In this case, f−f(a)+a is a holomorphic self-mapof S that fixes a and has derivative 1 at a, so it is the identity by the general formof Schwarz’s Lemma. Thus, from Corollary 6.5 f

′(a) = 1 implies f(z) = z + c

for some c ∈ R; the converse is trivial. If f′(a) = −1, then −f is a holomorphic

self-map of S with derivative 1 at a, and so f(z) = −z − c for some c ∈ R.

For a simply connected proper subregion Ω of C and a ∈ Ω, each hyperbolicdisk DΩ(a, r) = z ∈ Ω : dΩ(a, z) < r is simply connected and the closeddisk DΩ(a, r) is compact. When Ω is a disk or half-plane, hyperbolic disks areEuclidean disks since any conformal map of the unit disk onto a disk or half-planeis a Mobius transformation. Of course, this is no longer true when Ω is simplyconnected and not a disk or half-plane. For particular types of simply connectedregions, more can be said about hyperbolic disks than just the fact that they aresimply connected.

Theorem 7.11. Suppose Ω is a convex hyperbolic region. Then for any a ∈ Ωand all r > 0 the hyperbolic disc DΩ(a, r) is Euclidean convex.

Proof. Fix a ∈ Ω. Let h : D → Ω be a conformal mapping with h(0) = a. Sinceh(DD(0, r)) = DΩ(a, r), it suffices to show that h maps each disc DD(0, r) =D(0, tanh(r/2)) onto a convex set. Set R = tanh(r/2). Given b, c ∈ D(0, R)we must show (1 − t)h(b) + th(c) lies in h(D(0, R)) for t ∈ I. Choose S so that|b|, |c| < S < R and fix t ∈ I. The function

g(z) = (1 − t)h

(bz

S

)+ th

(cz

S

)

is holomorphic in D, g(0) = a and maps into Ω because Ω is convex. There-fore, f = h

−1 g is a holomorphic self-map of D that fixes the origin and sof(D(0, R)) ⊆ D(0, R). Then (1 − t)h(b) + th(c) = g(S) = h(f(S)) lies inh(D(0, S)) because f(S) ∈ D(0, S). Therefore, h(D(0, S)) = DΩ(a, r) is Eu-clidean convex.

This result is effectively due to Study who proved that if f is convex univalentin D, then for any Euclidean disk D contained in D, f(D) is Euclidean convex,see [13]. The converse of Theorem 7.11 is elementary: If Ω is a simply connectedproper subregion of C and there exists a ∈ Ω such that every hyperbolic diskDΩ(a, r) is Euclidean convex, then Ω is Euclidean convex since Ω = ∪DΩ(a, r) :r > 0, an increasing union of Euclidean convex sets. The radius of convexityfor a univalent function on D is 2−

√3; see [13]. This implies that if Ω is simply


connected, then for each a ∈ Ω and 0 < r < (1/2) log 3 the hyperbolic diskDΩ(a, r) is Euclidean convex.

In a general simply connected region hyperbolic geodesics are no longer arcsof circles or segments of lines. It is possible to give a simple geometric propertyof hyperbolic geodesics in Euclidean convex regions that characterize convexregions, see [19] and [20].

Exercises.

1. Let K = z = x + iy : x > 0. For a > 0 and r > 0 verify that theclosed hyperbolic disk DK(1, r) is the Euclidean disk with Euclidean centerc = cosh r and Euclidean radius R = sinh r. This Euclidean disk meets thereal axis at e

−r and er.

2. Suppose f : D → K is holomorphic. Prove that

(1 − |z|2)|f ′(z)| ≤ 2 Re f(z)

for all z ∈ D. When does equality hold?3. Suppose f : K → D is holomorphic. Prove that

2|f ′(z)|Re z ≤ 1 − |f(z)|2

for all z ∈ K. When does equality hold?4. Suppose Ω is a simply connected proper subregion of C that is (Euclidean)

starlike with respect to a ∈ Ω. This means that for each z ∈ Ω theEuclidean segment [a, z] is contained in Ω. For any r > 0 prove that thehyperbolic disk DΩ(a, r) is starlike with respect to a.

8. The Comparison Principle

There is a powerful, and very general, Comparison Principle for hyperbolicmetrics, which we state here only for simply connected plane regions. ThisPrinciple allows us to estimate the hyperbolic metric of a region in terms of otherhyperbolic metrics which are known, or which can be more easily estimated. Ingeneral it is not possible to explicitly calculate the density of the hyperbolicmetric, so estimates are useful.

Theorem 8.1 (Comparison Principle). Suppose that Ω1 and Ω2 are simply con-

nected proper subregions of C. If Ω1 ⊆ Ω2 then λΩ2≤ λΩ1

on Ω1. Further, if

λΩ1(z) = λΩ2

(z) at any point z of Ω2, then Ω1 = Ω2 and λΩ1= λΩ2

.

Proof. Let f(z) = z be the inclusion map of Ω1 into Ω2. Then the Schwarz-PickLemma gives λΩ2

(z) ≤ λΩ1(z). If equality holds at a point, then f is a conformal

bijection of Ω1 onto Ω2, that is, Ω1 = Ω2.

In other words, the Comparison Principle asserts that the hyperbolic metricon a simply connected region decreases as the region increases. The hyperbolicmetric on the disk Dr = z : |z| < r is 2r|dz|/(r2−|z|2) which decreases to zeroas r increases to +∞.


The Comparison Principle is used in the following way. Suppose that we wantto estimate the hyperbolic metric λΩ of a region Ω. We attempt to find regionsΩj with known hyperbolic metrics (or metrics that can be easily estimated) suchthat Ω1 ⊆ Ω ⊆ Ω2; then λΩ2

≤ λΩ ≤ λΩ1. The next result is probably the

simplest application of the Comparison Principle, and it gives an upper boundof the hyperbolic metric λΩ of a region Ω in terms of the Euclidean distance

d(z, ∂Ω) = inf|z − w| : w ∈ ∂Ωof z to the boundary of Ω. The geometric significance of this quantity is thatd(z, ∂Ω) is the radius of the largest open disk with center z that lies in Ω. Note,however, that d(z, ∂Ω) (which is sometimes denoted by δΩ(z) in the literature)is not conformally invariant. The metric

|dz|d(z, ∂Ω)

=|dz|

δΩ(z)

is called the quasihyperbolic metric on Ω. Example 7.2 shows that the quasihy-perbolic metric for a half-plane is the hyperbolic metric.

Theorem 8.2. Suppose that Ω is a simply connected proper subregion of C.

Then for all z ∈ Ω

(8.1) λΩ(z) ≤ 2

d(z, ∂Ω),

and equality holds if and only if Ω is a disk with center z.

Proof. Take any z0 in Ω, and let R = d(z0, ∂Ω) and D = z : |z − z0| < R. AsD ⊆ Ω the Comparison Principle and Example 7.1 yield

λΩ(z0) ≤ λD(z0) =2

R=

2

d(z0, ∂Ω),

which is (8.1). If λΩ(z0) = 2/d(z0, ∂Ω) then λΩ(z0) = λD(z0) so, by the Compar-ison Principle, Ω = D. The converse is trivial.

Theorem 8.2 gives an upper bound on the hyperbolic metric of Ω in terms ofthe Euclidean quantity d(z, ∂Ω). It is usually more difficult to obtain a lowerbound on the hyperbolic metric. For convex regions it is easy to use geometricmethods to obtain a good lower bound on the hyperbolic metric.

Theorem 8.3. Suppose that Ω is convex proper subregion of C. Then for all

z ∈ Ω

(8.2)1

d(z, ∂Ω)≤ λΩ(z)

and equality holds if and only if Ω is a half-plane.

Proof. We suppose that Ω is convex. Take any z in Ω and let ζ be one of thepoints on ∂Ω that is nearest to z. Let H be the supporting half-plane of Ω at ζ;


thus Ω ⊆ H, and the Euclidean line that bounds H is orthogonal to the segmentfrom z to ζ. Thus, from the Comparison Principle, for any z ∈ Ω

λΩ(z) ≥ λH(z) =1

|z − ζ| =1

d(z, ∂Ω)

which is (8.2). The equality statement follows from the Comparison Principleand Example 7.2.

Theorems 8.2 and 8.3 show that the hyperbolic and quasihyperbolic metricsare bi-Lipschitz equivalent on convex regions:

1

d(z, ∂Ω)≤ λΩ(z) ≤ 2

d(z, ∂Ω).

Lower bounds for the hyperbolic metric in terms of the quasihyperbolic metricare equivalent to covering theorems for univalent functions.

Theorem 8.4. Suppose that f is holomorphic and univalent in D, and that f(D)is a convex region. Then f(D) contains the Euclidean disk with center f(0) and

radius |f ′(0)|/2.

Proof. From Theorem 8.3 we have

2 = λD(0)

= λf(D)

(f(0)

)|f ′(0)|

≥ |f ′(0)|d

(f(0), ∂f(D)

) .

We deduce thatd

(f(0), ∂f(D)

)≥ |f ′(0)|/2,

so that f(D) contains the Euclidean disk with center f(0) and radius |f ′(0)|/2.

There is an analogous covering theorem for general univalent functions on theunit disk, see [13].

Theorem 8.5 (Koebe 1/4–Theorem). Suppose that f is holomorphic and uni-

valent in D. Then the region f(D) contains the open Euclidean disk with center

f(0) and radius |f ′(0)|/4.

The Koebe 1/4–Theorem gives a lower bound on the hyperbolic metric interms of the quasihyperbolic metric on a simply connected proper subregion ofC.

Theorem 8.6. Suppose that Ω is a simply connected proper subregion of C.

Then for all z ∈ Ω

(8.3)1

2d(z, ∂Ω)≤ λΩ(z)

and equality holds if and only if Ω is a slit-plane.


Proof. Fix z ∈ Ω and let f : D → Ω be a conformal map with f(0) = z. Koebe’s1/4–Theorem implies d(z, ∂Ω) ≥ |f ′(0)|/4. Now

2 = λΩ(f(0))|f ′(0)|≤ 4λΩ(z)d(z, ∂Ω)

which establishes (8.3). Sharpness follows from the sharp form of the Koebe1/4–Theorem and Example 7.7.

Theorems 8.2 and 8.6 show that the hyperbolic and quasihyperbolic metricsare bi-Lipschitz equivalent on simply connected regions:

(8.4)1

2d(z, ∂Ω)≤ λΩ(z) ≤ 2

d(z, ∂Ω).

Exercises.

1. (a) Suppose Ω is a simply connected proper subregion of C. Prove thatlimz→ζ λΩ(z) = +∞ for each boundary point ζ of Ω that lies in C.(b) Given an example of a simply connected proper subregion Ω of C thathas ∞ as a boundary point and λΩ(z) does not tend to infinity as z → ∞.

2. Suppose Ω is starlike with respect to the origin; that is, for each z ∈ Ω theEuclidean segment [0, z] is contained in Ω. Use the Comparison Theoremto prove that (8.3) holds; do not use Theorem 8.6.

9. Curvature and the Ahlfors Lemma

A conformal semimetric on a region Ω in C is ρ(z)|dz|, where ρ : Ω → [0, +∞)is a continuous function and z : ρ(z) = 0 is a discrete subset of Ω. A conformalsemimetric ρ(z)|dz| is a conformal metric if ρ(z) > 0 for all z ∈ Ω. The curvatureof a conformal semimetric ρ(z)|dz| can be defined at any point where ρ is positiveand of class C2.

Definition 9.1. Suppose ρ(z)|dz| is a conformal metric on a region Ω. If a ∈ Ω,ρ(a) > 0 and ρ(z) is of class C2 at a, then the Gaussian curvature of ρ(z)|dz| ata is

Kρ(a) = − log ρ(a)

ρ2(a),

where is the usual (Euclidean) Laplacian,

=∂

2

∂x2+

∂2

∂y2.

Typically, we shall speak of the curvature of a conformal metric rather thanGaussian curvature. In computing the Laplacian it is often convenient to use

= 4∂

2

∂z∂z= 4

∂2

∂z∂z,


where the complex partial derivatives are defined by

∂ =∂

∂z=

1

2

(∂

∂x− i

∂

∂y

),

∂ =∂

∂z=

1

2

(∂

∂x+ i

∂

∂y

).

Alternatively the Laplacian expressed can be expressed in terms of polar coordi-nates, namely

=∂

2

∂r2+

1

r

∂

∂r+

1

r2

∂2

∂θ2.

One reason semimetrics play an important role in complex analysis is thatthey transform simply under holomorphic functions.

Definition 9.2. Suppose ρ(w)|dw| is a semimetric on a region Ω and f : ∆ → Ωis a holomorphic function. The pull-back of ρ(w)|dw| by f is

(9.1) f∗(ρ(w)|dw|) = ρ(f(z))|f ′(z)||dz|.

Since (ρ f)|f ′| is a continuous nonnegative function defined on ∆, the pull-back f

∗(ρ(w)|dw|) of ρ(w)|dw| is a semimetric on ∆ provided f is nonconstant.Sometimes we write simply f

∗(ρ) to denote the pull-back. However, the nota-tion (9.1) is precise and indicates that the formal substitution w = f(z) convertsρ(w)|dw| into f

∗(ρ(w)|dw|). The pull-back operation has several useful proper-ties:

(f g)∗(ρ(w)|dw|) = g∗ (f ∗(ρ(w)|dw|))

and(f−1)∗ = (f ∗)−1

,

when f is a conformal mapping. If f : Ω1 → Ω2 is a conformal mapping ofsimply connected proper subregions of C, then the conclusion of Theorem 6.3 inthe pull-back notation is: f

∗(λΩ2) = λΩ1

.

In the context of complex analysis, a fundamental property of the curvatureis its conformal invariance. More generally, curvature is invariant under thepull-back operation.

Theorem 9.3. Suppose Ω and ∆ are regions in C, ρ(w)|dw| is a metric on Ω and

f : ∆ → Ω is a holomorphic function. Suppose a ∈ ∆, f′(a) 6= 0, ρ(f(a)) > 0

and ρ is of class C2 at f(a). Then Kf∗(ρ)(a) = Kρ(f(a)).

Proof. Recall that f∗(ρ(w)|dw|) = ρ(f(z))|f ′(z)||dz|. Now,

log (ρ(f(z))|f ′(z)|) = log ρ(f(z)) + log |f ′(z)|

= log ρ(f(z)) +1

2log f

′(z) +1

2log f ′(z),

so that∂

∂zlog (ρ(f(z))|f ′(z)|) =

∂ log ρ

∂w(f(z))f ′(z) +

1

2

f′′(z)

f ′(z).


Then

∂2

∂z∂zlog (ρ(f(z))|f ′(z)|) =

∂2 log ρ

∂w∂w(f(z))f ′(z)f ′(z)

=∂

2 log ρ

∂w∂w(f(z))|f ′(z)|2

gives

z[log (ρ(f(z))|f ′(z)|)] = (w log ρ) (f(z))|f ′(z)|2.This is the transformation law for the Laplacian under a holomorphic function.Consequently,

Kf∗(ρ)(a) = −z log(ρ(f(a))|f ′(a)|)ρ2(f(a))|f ′(a)|2

= −(w log ρ) (f(a))|f ′(a)|2ρ2(f(a))|f ′(a)|2

= Kρ(f(a)).

Theorem 9.4. The hyperbolic metric on a simply connected proper subregion of

C has constant curvature −1.

Proof. First, we establish the result for the unit disk. From

λD(z) =2

1 − |z|2 =2

1 − zz

we obtain

∂2

∂z∂zlog

2

1 − zz= − ∂

2

∂z∂zlog(1 − zz)

=∂

∂z

z

1 − zz

=1

(1 − zz)2.

Consequently, KλD(z) = −1.

The general case of the hyperbolic metric on a simply connected proper sub-region Ω of C follows from Theorem 9.3 since f

∗(λD(w)|dw|) = λΩ(z)|dz| for anyconformal map f : Ω → D.

Ahlfors recognized that the Schwarz-Pick Lemma was a consequence of anextremely important maximality property of the hyperbolic metric in D.

Theorem 9.5 (Maximality of the hyperbolic metric). Suppose ρ(z)|dz| is a C2

semimetric on a simply connected proper subregion Ω of C such that Kρ(z) ≤ −1whenever ρ(z) > 0. Then ρ ≤ λΩ on Ω.


Proof. First, we assume Ω = D. Given z0 in D choose any r satisfying |z0| <

r < 1. The hyperbolic metric on the disk Dr = z : |z| < r is

λr(z) =2r

r2 − |z|2 .

Consider the function

v(z) =ρ(z)

λr(z)

which is defined on the disk Dr. Then v(z) ≥ 0 when |z| < r, and v(z) → 0 as|z| → r, so that v attains its maximum at some point a in Dr. It suffices to showthat v(a) ≤ 1 for then v(z) ≤ 1 on Dr and we have

ρ(z0) ≤2r

r2 − |z0|2.

By letting r → 1 we find that ρ(z0) ≤ λD(z0).

We now show that v(a) ≤ 1. If ρ(a) = 0, then v(a) = 0 < 1. Otherwise,ρ(a) > 0 and Kρ(a) ≤ −1. As a is a local maximum of v, it is also a localmaximum of log v so that

∂2 log v

∂x2(a) ≤ 0,

∂2 log v

∂y2(a) ≤ 0.

We deduce that

0 ≥( log v

)(a)

=( log ρ

)(a) −

( log λr

)(a)

= −Kρ(a)ρ(a)2 + Kλr(a)λr(a)2

≥ ρ(a)2 − λr(a)2.(9.2)

This implies that v(a) ≤ 1, and completes the proof in the special case Ω = D.

We now turn to the general case. Let h : D → Ω be a conformal mapping.Then h

∗(ρ(w)|dw|) := τ(z)|dz| is a C2 semimetric on D such that Kτ (z) ≤ −1whenever τ(z) > 0. Hence,

ρ(h(z))|h′(z)| ≤ λD(z) = λΩ(h(z))|h′(z)|,and so ρ ≤ λΩ on Ω.

In fact, Ahlfors actually established a more general result (see [1] and [2]). Thestronger conclusion that either ρ < λΩ or else ρ = λΩ is valid but less elementary.This sharp result was established by Heins [15]. Simpler proofs of the strongerconclusion are due to Chen [12], Minda [28] and Royden [32].

The Schwarz-Pick Lemma is a special case of Theorem 9.5. If f : Ω1 → Ω2

is a nonconstant holomorphic function, then f∗(λΩ2

(w)|dw|) is a semimetric onΩ1 with curvature −1 at each point where f

′ is nonvanishing, so is dominatedby the hyperbolic metric λΩ1

(z)|dz|, or equivalently, (6.4) holds. The equalitystatement associated with (6.4) follows from the sharp version of Theorem 9.5.


Theorem 9.6. There does not exist a C2 semimetric ρ(z)|dz| on C such that

Kρ(z) ≤ −1 whenever ρ(z) > 0.

Proof. Suppose there existed a semimetric ρ(z)|dz| on C such that Kρ(z) ≤ −1whenever ρ(z) > 0. Theorem 9.5 applied to the restriction of this metric to thedisk z : |z| < r gives

(9.3) ρ(z) ≤ λr(z) =2r

r2 − |z|2for |z| < r. If we fix z and let r → +∞, (9.3) gives ρ(z) = 0 for all z ∈ C. Thiscontradicts the fact that a semimetric vanishes only on a discrete set.

Corollary 9.7 (Liouville’s Theorem). A bounded entire function is constant.

Proof. Suppose f is a bounded entire function. There is no harm in assumingthat |f(z)| < 1 for all z ∈ C. If f were nonconstant, then f

∗(λD(z)|dz|) wouldbe a semimetric on C with curvature at most −1, a contradiction.

Theorem 9.3 provides a method to produce metrics with constant curvature−1. Loosely speaking, bounded holomorphic functions correspond to metricswith curvature −1. If f : Ω → D is holomorphic and locally univalent (f ′ doesnot vanish), then f

∗(λD(z)|dz|) has curvature −1 on Ω. In fact, on a simplyconnected proper subregion of C every metric with curvature −1 has this form;see [36]. This reference also contains a stronger result that represents certainsemimetrics with curvature −1 at points where the semimetric is nonvanishingby holomorphic (not necessarily locally univalent) maps of Ω into D.

Theorem 9.8 (Representation of Negatively Curved Metrics). Let ρ(z)|dz| be a

C3 conformal metric on a simply connected proper subregion Ω of C with constant

curvature −1. Then ρ(z)|dz| = f∗(λD(w)|dw|) for some locally univalent holo-

morphic function f : Ω → D. The function f is unique up to post-composition

with an isometry of the hyperbolic metric. Given a ∈ Ω the function f represent-

ing the metric is unique if f is normalized by f(a) = 0 and f′(a) > 0.

Moreover, ρ(z)|dz| = f∗(λD(w)|dw|) is complete if and only if f is a conformal

bijection; that is, the hyperbolic metric is the only conformal metric on Ω thathas curvature −1 and is complete.

Exercises.

1. Determine the curvature of the Euclidean metric |dz| and of the sphericalmetric σ(z)|dz| = 2|dz|/(1 + |z|2).

2. Show that (1 + |z|2)|dz| has negative curvature on C.3. Determine the curvature of e

x|dz| on C.4. Determine the curvature of |dz|/|z| on C \ 0.5. Prove there does not exist a semimetric on C \ 0 with curvature at most

−1.6. Prove the following extension of Liouville’s Theorem: If f is an entire

function and f(C) ⊆ Ω, where Ω is a simply connected proper subregionof C, then f is constant.


10. The hyperbolic metric on a hyperbolic region

In order to transfer the hyperbolic metric from the unit disk to nonsimplyconnected regions, a substitute for the Riemann Mapping Theorem is needed. Forthis reason we must understand holomorphic coverings. For the general theoryof topological covering spaces the reader should consult [22]. A holomorphicfunction f : ∆ → Ω is called a covering if each point b ∈ Ω has an openneighborhood V such that f

−1(V ) = ∪Uα : α ∈ A is a disjoint union of opensets Uα such that f |Uα, the restriction of f to Uα, is a conformal map of Uα ontoV . Trivially, a conformal mapping f : ∆ → Ω is a holomorphic covering. If Ω issimply connected, then the only holomorphic coverings f : ∆ → Ω are conformalmaps of a simply connected region ∆ onto Ω.

Example 10.1. The complex exponential function exp : C → C \ 0 is aholomorphic covering. Consider any w ∈ C \ 0 and let θ = arg w be anyargument for w. Let V = C \ −re

iθ : r ≥ 0 be the complex plane slit from theorigin along the ray opposite from w. Then exp−1(V ) =

⋃Sn : n ∈ Z, whereSn = z : θ − nπ < Im z < θ + nπ. Note that exp maps each horizontal stripSn of width 2π conformally onto V .

A region Ω is called hyperbolic provided C∞ \Ω contains at least three points.The unit disk covers every hyperbolic plane region; that is, there is a holomorphiccovering h : D → Ω for any hyperbolic region Ω. As a consequence we demon-strate that every hyperbolic region has a hyperbolic metric that is real-analyticwith constant curvature −1.

Theorem 10.2 (Planar Uniformization Theorem). Suppose Ω is a region in C.

There exists a holomorphic covering f : D → Ω if and only if Ω is a hyperbolic

region. Moreover, if a ∈ Ω, then there is a unique holomorphic universal covering

h : D → Ω with h(0) = a and h′(0) > 0.

For a proof of the Planar Uniformization see [14] or [34]. If Ω is a simplyconnected hyperbolic region, then any holomorphic universal covering h : D →Ω is a conformal mapping. Therefore, the Riemann Mapping Theorem is aconsequence of the Planar Uniformization Theorem. When Ω is a nonsimplyconnected hyperbolic region, then a holomorphic covering h : D → Ω is neverinjective. In fact, for each a ∈ Ω, h

−1(a) is an infinite discrete subset of D. Ifh : D → Ω is one holomorphic universal covering, then hg : g ∈ A(D) is the setof all holomorphic universal coverings of D onto Ω. The Planar UniformizationTheorem enables us to project the hyperbolic metric from the unit disk to anyhyperbolic region.

Theorem 10.3. Given a hyperbolic region Ω there is a unique metric λΩ(w)|dw|on Ω such that h

∗(λΩ(w)|dw|) = λD(z)|dz| for any holomorphic universal cover-

ing h : D → Ω. The metric λΩ(w)|dw| is real-analytic with curvature −1.

Proof. We construct a metric with curvature −1 on any hyperbolic region. Firstwe define the metric locally. For a hyperbolic region Ω, let h : D → Ω be


a holomorphic covering. A metric is defined on Ω as follows. For any simplyconnected subregion U of Ω let H = h

−1 denote a branch of the inverse thatis holomorphic on U . Set λΩ(z) = λD(H(z))|H ′(z)|. This defines a metric withcurvature −1 on U . In fact, this defines a metric on Ω. Suppose U1 and U2 aresimply connected subregions of Ω and U1 ∩ U2 is nonempty. Let Hj be a single-valued holomorphic branch of h

−1 defined on Uj. Then there is a g ∈ A(D) suchthat H2 = g H1 locally on U1 ∩ U2. Hence,

H∗2(λD(z)|dz|) = (g H1)

∗(λD(z)|dz|)= H

∗1(g∗(λD(z)|dz|))

= H∗1(λD(z)|dz|)

since each conformal automorphism of D is an isometry of the hyperbolic metricλD(z)|dz|. Therefore, λΩ(z) is defined independently of the branch of h

−1 thatis used and h

∗(λΩ) = λD.

Moreover, this metric is independent of the covering. Suppose k : D → Ω isanother covering. Then k = h g for some g ∈ A(D), and so

k∗(λΩ) = (h g)∗(λΩ)

= g∗(h∗(λΩ))

= g∗(λD)

= λD.

That λΩ is real-analytic is clear from its construction. That the curvature is −1follows from h

∗(λΩ) = λD and Theorems 9.3 and 9.4.

The unique metric λΩ(w)|dw| on a hyperbolic region Ω given by Theorem 10.3is called the hyperbolic metric on Ω. The hyperbolic distance on a hyperbolicregion is complete. The hyperbolic distance dΩ is defined by

dΩ(z, w) = inf ℓΩ(γ),

where the infimum is taken over all piecewise smooth paths γ in Ω that joiningz and w. Unlike the case of simply connected regions, a holomorphic coveringf : D → Ω onto a multiply connected hyperbolic region is not an isometry, butonly a local isometry. That is, each point a ∈ Ω has a neighborhood U such thatf |U is an isometry. In general, one can only assert that dΩ(f(z), f(w)) ≤ dD(z, w)for z, w ∈ D. When Ω is multiply connected, then f is not injective, so thereexist distinct z, w ∈ D with f(z) = f(w). In this situation, dΩ(f(z), f(w)) = 0 <

dD(z, w).

In general, the hyperbolic metric is not just invariant under conformal map-pings, but is invariant under holomorphic coverings.

Theorem 10.4 (Covering Invariance). If f : ∆ → Ω is a holomorphic covering

of hyperbolic regions, then f∗(λΩ(w)|dw|) = λ∆(z)|dz|. In other words, every

holomorphic covering of hyperbolic regions is a local isometry.


Proof. Let h : D → ∆ be a holomorphic covering. Then k = f h : D → Ω isalso a holomorphic covering, so

λD = k∗(λΩ)

= h∗(f ∗(λΩ)).

Thus, f∗(λΩ) is a conformal metric on ∆ whose pull-back to the unit disk by a

covering projection is λD, so f∗(λΩ) is the hyperbolic metric on ∆ by Theorem

10.3.

Theorem 10.4 implies that every h ∈ A(Ω) is an isometry of the hyperbolicmetric, and more generally, each holomorphic self-covering h of Ω is a localisometry of the hyperbolic metric. A hyperbolic region can have self-coveringsthat are not conformal automorphisms, see Section 12.

The maximality property of the hyperbolic metric given in Theorem 9.5 re-mains valid for hyperbolic regions. As noted after the proof of Theorem 9.5 thismeans that a version of the Schwarz-Pick Lemma holds for holomorphic mapsbetween hyperbolic regions. In order to establish a sharp result, we provide anindependent proof.

Theorem 10.5 (Schwarz-Pick Lemma - general version). Suppose ∆ and Ω are

hyperbolic regions and f : ∆ → Ω is holomorphic. Then for all z ∈ ∆,

(10.1) λΩ(f(z))|f ′(z)| ≤ λ∆(z).

If f : ∆ → Ω is a covering projection, then λΩ(f(z))|f ′(z)| = λ∆(z) for all

z ∈ ∆. If there exists a point in ∆ such that equality holds in (10.1), then f is

a covering.

Proof. Let k : D → ∆ and h : D → Ω be holomorphic coverings. The functionf k : D → Ω lifts relative to h to a holomorphic function F : D → D. Thenf k = h F and the Schwarz-Pick Lemma for the unit disk give

k∗(f ∗(λΩ)) = (f k)∗(λΩ)

= (h F )(λΩ)

= F∗(h∗(λΩ))

= F∗(λD)

≤ λD

= k∗(λ∆).

Because k is a surjective local homeomorphism, the inequality k∗(f ∗(λΩ)) ≤

k∗(λ∆) gives the inequality (10.1). Because h and k are coverings, f is a covering

if and only if F is a covering. This observation establishes the sharpness.

We need to establish a result about holomorphic self-coverings of a hyperbolicregion that have a fixed point in order to obtain a good analog of Schwarz’sLemma for hyperbolic regions.


Theorem 10.6. A self-covering of a hyperbolic region that fixes a point is a

conformal automorphism. In particular, if a hyperbolic region Ω is not simply

connected and a ∈ Ω, then A(Ω, a) is isomorphic to the group of nth roots of

unity for some positive integer n.

Proof. Suppose Ω is a hyperbolic region, a ∈ Ω and f is a self-covering of Ωthat fixes a. We prove f is a conformal automorphism. The result is trivialif Ω is simply connected since every covering of a simply connected region isa homeomorphism. Let h : D → Ω be a holomorphic universal covering withh(0) = a and h

′(0) > 0. Because a covering is surjective, it suffices to prove

f is injective. Let f be the lift of f h relative to h that satisfies f(0) = 0.

Since h and f h are coverings, so is f . Because D is simply connected, f is aconformal automorphism of D. Then f(z) = e

iθz for some θ ∈ R. Because Ω

is not simply connected, the fiber h−1(a) contains infinitely many points besides

0. As this fiber is a discrete subset of D, the nonzero elements of h−1(a) have a

minimum positive modulus r; say h−1(a) ∩ z : |z| = r = aj : j = 1, . . . ,m.

From f(h−1(a)) = h−1(a), we conclude that f induces a permutation of the set

aj : j = 1, . . . ,m. Therefore, there exists n ≤ m! such that fn is the identity.

Then fnh = h f

n = h and so fn is the identity. If n = 1, then f is the identity.

If n ≥ 2, then fn = I, the identity, implies f is a conformal automorphism of Ω

with inverse fn−1.

This argument shows that if Ω is not simply connected, then there is a non-negative integer m such that for all f ∈ A(Ω, a), f

m is the identity. Therefore,f′(a)m = 1, or f

′(a) is an mth root of unity. Thus, f 7→ f′(a) defines a homo-

morphism of A(Ω, a) into the unit circle T and the image is a subgroup of themth roots of unity. Hence, A(Ω, a) is a finite group isomorphic to the group ofnth roots of unity for some positive integer n.

Example 10.7. Let AR = z : 1/R < |z| < R, where R > 1. The groupA(AR, 1) has order two; the only conformal automorphisms of AR that fix 1 arethe identity map and f(z) = 1/z.

Corollary 10.8 (Schwarz’s Lemma - General Version). Suppose Ω is a hyper-

bolic region, a ∈ Ω and f is a holomorphic self-map of Ω that fixes a. Then

|f ′(a)| ≤ 1 and equality holds if and only if f ∈ A(Ω, a), the group of confor-

mal automorphisms of Ω that fix a. Moreover, f′(a) = 1 if and only if f is the

identity.

Proof. The Schwarz-Pick Lemma implies |f ′(a)| ≤ 1 with equality if and only iff is a self-covering of Ω that fixes a. Each f ∈ A(Ω, a) is a covering, so |f ′(a)| = 1.If f is a self-covering of Ω that fixes a, then f ∈ A(Ω, a) by Theorem 10.6. Weuse the proof of Theorem 10.6 to verify that f

′(a) = 1 implies f is the identity.

Let f be the lift of f h as in the proof of Theorem 10.6. Then h f = f h

gives 1 = f′(a) = f

′(0), so f is the identity. This implies f is the identity.

Picard established a vast generalization of Liouville’s Theorem.


Theorem 10.9 (Picard’s Small Theorem). If an entire function omits two finite

complex values, then f is constant.

Proof. Suppose f is an entire function and f(C) ⊆ C\a, b := Ca,b, where a andb are distinct complex numbers. We derive a contradiction if f were nonconstant.The region Ca,b is hyperbolic; let λa,b(z)|dz| denote the hyperbolic metric on Ca,b.If f were nonconstant, then f

∗(λa,b(z)|dz|) would be a semi-metric on C withcurvature at most −1; this contradicts Theorem 9.6.

Exercises.

1. Verify that f(z) = exp(iz) is a covering of the upper half-plane H onto thepunctured disk D \ 0.

2. Verify that for each nonzero integer n the function pn(z) = zn defines a

holomorphic covering of the punctured plane C \ 0 onto itself.

3. Verify that for each positive integer n the function pn(z) = zn defines a

holomorphic covering of the punctured disk D \ 0 onto itself.

4. Suppose Ω is a hyperbolic region and a ∈ Ω. Let F denote the family ofall holomorphic functions f : D → Ω such that f(0) = a and set M =sup|f ′(0)| : f ∈ F. Prove M is finite and for f ∈ F , |f ′(0)| = M

if and only if f is a holomorphic covering of D onto Ω. Conclude thatM = 2/λΩ(a).

5. Suppose Ω is a hyperbolic region in C and a, b ∈ Ω are distinct points. Iff is a holomorphic self-map of Ω that fixes a and b, prove f is a conformalautomorphism of Ω with finite order. Give an example to show that f neednot be the identity when Ω is not simply connected.

11. Hyperbolic distortion

In Section 5 the hyperbolic distortion of a holomorphic self-map of the unitdisk was introduced. We now define an analogous concept for holomorphic mapsof hyperbolic regions.

Definition 11.1. Suppose Ω and ∆ are hyperbolic regions and f : ∆ → Ω isholomorphic. The (local) hyperbolic distortion factor for f at z is

f∆,Ω(z) :=

λΩ(f(z))|f ′(z)|λ∆(z)

= limw→z

dΩ(f(z), f(w))

d∆(z, w).

If Ω = ∆, write f∆ in place of f

∆,Ω.

The hyperbolic distortion factor defines a mapping of ∆ into the closed unitdisk by the Schwarz-Pick Lemma. If f is not a covering, then the hyperbolicdistortion factor gives a map of ∆ into the unit disk. There is a Schwarz-Picktype of result for the hyperbolic distortion factor which extends Corollary 5.7 toholomorphic maps between hyperbolic regions.


Theorem 11.2 (Schwarz-Pick Lemma for Hyperbolic Distortion). Suppose ∆and Ω are hyperbolic regions and f : ∆ → Ω is holomorphic. If f is not a

covering, then

(11.1) dD(f∆,Ω(z), f∆,Ω(w)) ≤ 2d∆(z, w)

for all z, w ∈ ∆.

Proof. Fix w ∈ Ω. Let h : D → ∆ and k : D → Ω be holomorphic coveringswith h(0) = w and k(0) = f(w). Then there is a lift of f to a self-map f of D

such that k f = f h. f is not a conformal automorphism of D because f isnot a covering of ∆ onto Ω. We begin by showing that f

D(z) = f∆,Ω(h(z)) for

all z in D. From k f = f h and λD(z) = λ∆(h(z))|h′(z)| we obtain

f∆,Ω(h(z)) =

λΩ(f(h(z)))|f ′(h(z))|λ∆(h(z))

=λΩ(k(f(z)))|k′(f(z))||f ′(z)|

λ∆(h(z))|h′(z)|

=λD(f(z))|f ′(z)|

λD(z)

= fD(z).

Now we establish (11.1). For z ∈ Ω there exists z ∈ h−1(z) with dD(0, z) =

dΩ(w, z). Then

dD(f∆,Ω(z), f∆,Ω(w)) = dD(f∆,Ω(h(z)), f∆,Ω(h(0)))

= dD(fD(z), fD(0))

≤ 2dD(z, 0)

= 2d∆(z, w).

Corollary 11.3. Suppose ∆ and Ω are hyperbolic regions. Then for any holo-

morphic function f : ∆ → Ω,

(11.2) f∆,Ω(z) ≤ f

∆,Ω(w) + tanh d∆(z, w)

1 + f∆,Ω(w) tanh d∆(z, w).

for all z, w ∈ ∆.

Proof. Inequality (11.2) is trivial when f is a covering since both sides areidentically one, Thus, it suffices to establish the inequality when f is not acovering of ∆ onto Ω. Then

dD(0, f∆,Ω(z)) ≤ dD(0, f∆,Ω(w)) + dD(f∆,Ω(z), f∆,Ω(w))

≤ dD(0, f∆,Ω(w)) + 2d∆(z, w)


gives

f∆,Ω(z) = tanh

(1

2dD(0, f∆,Ω(z))

)

≤ tanh

(1

2dD(0, f∆,Ω(w)) + d∆(z, w)

)

=f

∆,Ω(w) + tanh d∆(z, w)

1 + f∆,Ω(w) tanh d∆(z, w).

Exercises.

1. For a holomorphic function f : D → K, explicitly calculate fD,K(z).

2. Suppose Ω is a simply connected proper subregion of C and a ∈ Ω. LetH(Ω, a) denote the set of holomorphic self-maps of Ω that fix a. Prove thatfΩ(a) : f ∈ H(Ω, a) is the closed unit interval [0, 1].

12. The hyperbolic metric on a doubly connected region

There is a simple conformal classification of doubly connected regions in C∞.If Ω is a doubly connected region in C∞, then Ω is conformally equivalent toexactly one of:

(a) C∗ = C \ 0,

(b) D∗ = D \ 0, or

(c) an annulus A(r, R) = z : r < |z| < R, where 0 < r < R.

In the first case Ω itself is the extended plane C∞ punctured at two points andso is not hyperbolic. In this section we calculate the hyperbolic metric for thepunctured unit disk D

∗ and for the annulus AR = z : 1/R < |z| < R, whereR > 1.

12.1. Hyperbolic metric on the punctured unit disk. To determine thehyperbolic metric on D

∗ we make use of a holomorphic covering from H onto D∗

and Theorem 10.4. The function h(z) = exp(iz) is a holomorphic covering fromH onto D

∗. Therefore, the density of the hyperbolic metric on D∗ is

λD∗(z) =1

|z| log(1/|z|) .

For simply connected hyperbolic regions the only hyperbolic isometries of thehyperbolic metric are conformal automorphisms. For multiply connected re-gions there can be self-coverings that leave the hyperbolic metric invariant. Forinstance, the maps z 7→ z

n, n = 2, 3, . . ., are self-coverings of D∗ that leave

λD∗(z)|dz| invariant. Up to composition with a rotation about the origin theseare the only self-coverings of D

∗ that are not automorphisms.

Because each hyperbolic geodesic in D∗ is the image of a hyperbolic geodesic

in H under h, every radial segment [reiθ, Re

iθ], where 0 < r < R < 1, is part of


a hyperbolic geodesic. Since the density λD∗ is independent of θ, the hyperboliclength of a geodesic segment σr,R = [reiθ

, Reiθ] is independent of θ. Direct

calculation gives

ℓD∗(σr,R) =

∫

[r,R]

λD∗(z)|dz| =

∫ R

r

dt

t log t= log

∣∣∣∣log R

log r

∣∣∣∣ .

As the formula shows, this length tends to infinity if either r → 0 or R → 1which also follows from the completeness of the hyperbolic metric. The Euclideancircle Cr = z : |z| = r, where 0 < r < 1, is not a hyperbolic geodesic; it hashyperbolic length

ℓD∗(Cr) =

∫

|z|=r

|dz||z| log(1/|z|) =

2π

log(1/r).

The hyperbolic length of Cr approaches 0 when r → 0 and ∞ when r → 1. Thehyperbolic area of the annulus A(r, R) = z : r < |z| < R ⊂ D

∗ is

aD∗(A(r, R)) =

∫ ∫

A(r,R)

1

|z|2 log2 |z| dx dy

= 2π

∫ R

r

dt

t log2t

= 2π

(1

log(1/R)− 1

log(1/r)

).

The hyperbolic area of A(r, R) tends to infinity when R → 1 and has the finitelimit 2π/ log(1/R) when r → 0.

There is a Euclidean surface in R3 that is isometric to z : 0 < |z| < 1/e

with the restriction of the hyperbolic metric on D∗ and makes it easy to see

these curious results about length and area in a neighborhood of the puncture.If a tractrix is rotated about the y-axis and the resulting surface is given thegeometry induced from the Euclidean metric on R

3, then this surface has constantcurvature −1 and is isometric to z : 0 < |z| < 1/e with the restriction of thehyperbolic metric on D

∗. This picture provides a simple isometric embedding ofa portion of D

∗ into R3. Radial segments correspond to rotated copies of the

tractrix and these have infinite Euclidean length. At the same time the surfacehas finite Euclidean area.

Recall that for a simply connected region Ω, the hyperbolic density λΩ andthe quasihyperbolic density 1/δΩ are bi-Lipschitz equivalent; see (8.4). Thesetwo metrics are not bi-Lipschitz equivalent on D

∗ because the behavior of λD∗

near the unit circle differs from its behavior near the origin. For 1/2 < |z| < 1,δD∗(z) = 1 − |z| and so

lim|z|→1

δD∗(z)λD∗(z) = 1.

For 0 < |z| < 1/2, δ(z) = |z| and so

lim|z|→1

δD∗(z)λD∗(z) = 0.


12.2. Hyperbolic metric on an annulus. Now we determine the hyperbolicmetric on an annulus by using a holomorphic covering from a strip onto anannulus. In each conformal equivalence class of annuli we choose the uniquerepresentative that is symmetric about the unit circle.

For 0 < r < R let A(r, R) = z : r < |z| < R. The number mod(A(r, R)) =log(R/r) is called the modulus of A(r, R). Two annuli A(rj, Rj), j = 1, 2, areconformally (actually Mobius) equivalent if and only if R1/r1 = R2/r2; that is, if

and only if they have equal moduli. If S =√

(R/r), then AS = z : 1/S < |z| <

S is the unique annulus conformally equivalent to A(r, R) that is symmetricabout the unit circle. Here symmetry means AS is invariant under z 7→ 1/z,reflection about the unit circle. Note that mod(AS) = 2 log S.

The function k(z) = exp(z) is a holomorphic universal covering from thevertical strip Sb = z : |Im z| < b, where b = log R, onto the annulus AR = z :1/R < |z| < R. Therefore, the density of the hyperbolic metric of the annulusAR is

λR(z) =π

2 log R

1

|z| cos(

π log |z|2 log R

) .

Example 12.1. We investigate the hyperbolic lengths of the Euclidean circlesCr = z : |z| = r in AR. The hyperbolic length of Cr is

ℓR(Cr) =

∫

|z|=r

π

2 log R

|dz||z| cos

(π log |z|2 log R

) =π

2

(log R) cos(

π log r

2 log R

) .

The symmetry of AR about the unit circle is reflected by the fact that twocircles symmetric about the unit circle have the same hyperbolic length. Also,the hyperbolic length of Cr increases from π

2/ log R = 2π2

/mod AR to ∞ as r

increases from 1 to R. Hence, the hyperbolic lengths of the Euclidean circles Cr

in AR have a positive minimum hyperbolic length. The Euclidean circle C1 is ahyperbolic geodesic; Cr is not a hyperbolic geodesic when r 6= 1.

If γn(t) = exp(2πint), then I(γn, 0) = n, where I(δ, 0) denotes the index orwinding number of a closed path δ about the origin, and

ℓR(γn) =2π2|n|

mod A(R).

We now show that γn has minimal hyperbolic length among all closed paths inAR that wind n times about the origin.

Theorem 12.2. Suppose γ is a piecewise smooth closed path in AR and I(γ, 0) =n 6= 0. Then

(12.1)2π2|n|

mod A(R)≤ ℓR(γ),

where ℓR(γ) denotes the hyperbolic length of γ. Equality holds in (12.1) if and

only if γ is a monotonic parametrization of the unit circle traversed n times.


Proof. Suppose γ : [0, 1] → AR is a closed path with I(γ, 0) = n 6= 0. Then

ℓR(γ) =

∫

γ

λR(z) |dz|

=π

2 log R

∫1

0

|γ′(t)| dt

|γ(t)| cos(

π log |γ(t)|2 log R

)

≥ π

2 log R

∫1

0

|γ′(t)||γ(t)|

since

(12.2) 0 < cos

(π log |γ(t)|

2 log R

)≤ 1

and equality holds if and only if |γ(t)| = 1 for t ∈ [0, 1]. Next,∫

1

0

|γ′(t)||γ(t)| ≥

∣∣∣∣∫

1

0

γ′(t)

γ(t)dt

∣∣∣∣(12.3)

=

∣∣∣∣∫

γ

dz

z

∣∣∣∣

= |2πiI(γ, 0)|= 2π|n|.

Hence,

ℓR(γ) ≥ π2|n|

log R=

2π2|n|mod AR

.

It is straightforward to verify that if γn(t) = exp(2πint), t ∈ [0, 1], then equalityholds in (12.1). Conversely, suppose γ is a path for which equality holds. Thenequality holds in (12.2), so |γ(t)| = 1 for t ∈ [0, 1]. Let δ : [0, 1] → C be alift of γ relative to the covering exp : C → C

∗. From I(γ, 0) = n, we obtainδ(1) − δ(0) = 2πni. The condition |γ(t)| = 1 implies δ(t) ∈ iR for t ∈ [0, 1].The function h(t) = (δ(t) − δ(0))/2πi is real-valued, h(0) = 0 and h(1) = n.Also, γ(t) = exp(2πih(t) + δ(0)). Equality must hold in (12.3) and this meansγ′(t)/γ(t) = 2πih

′(t) has constant argument. Hence, h′(t) is either positive

or negative, so t 7→ exp(2πih(t) + δ(0) is a parametrization of the unit circlestarting at γ(0) = exp δ(0) and the unit circle is traversed either clockwise orcounterclockwise.

Exercises.

1. Consider the metric |dz|/|z| on C \ 0 := C∗ and let ℓC∗(γ) denote the

length of a path γ in C∗ relative to this metric. If γ is a closed path in C

∗,prove

ℓC∗(γ) ≥ 2π|I(γ, 0)|.When does equality hold?

2. Suppose f is holomorphic on D and f(D) ⊆ D \ 0. Prove that |f ′(0)| ≤2/e. Determine when equality holds.


13. Rigidity theorems

If a holomorphic mapping of hyperbolic regions is not a covering, then strictinequality holds in the general version of the Schwarz-Pick Lemma. It is oftenpossible to provide a quantitative version of this strict inequality that is inde-pendent of the holomorphic mapping for multiply connected regions. We beginby establishing a refinement of Schwarz’s Lemma.

Lemma 13.1. Suppose 0 6= a ∈ D and 0 < t < 1. If f is a holomorphic self-map

of D, f(0) = 0 and |f(a)| ≤ t|a|, then

(13.1) |f ′(0)| ≤ t + |a|1 + t|a| < 1.

Proof. The Three-point Schwarz-Pick Lemma (Theorem 4.4) with z = 0 = v

and w = a gives

dD(f ′(0), f(a)/a) = dD(f ∗(0, 0), f ∗(a, 0))

≤ dD(0, a).

Hence,

dD(0, |f ′(0)|) = dD(0, f ′(0))

≤ dD(0.f(a)/a) + dD(0, a)

≤ dD(0, t) + dD(0,−|a|)= d(−|a|, t),

which is equivalent to (13.1).

Theorem 13.2. Suppose ∆ and Ω are hyperbolic regions with a ∈ ∆, b ∈ Ωand ∆ is not simply connected. There is a constant α = α(a, ∆; b, Ω) ∈ [0, 1)such that if f : ∆ → Ω is any holomorphic mapping with f(a) = b that is not a

covering, then

(13.2) λΩ(f(a))|f ′(a)| ≤ αλ∆(a);

or equivalently, f∆,Ω(a) ≤ α. Moreover, for all z ∈ ∆

(13.3) f∆,Ω(z) ≤ α + tanh d∆(a, z)

1 + α tanh d∆(a, z)< 1.

Proof. Let h : D → ∆ and k : D → Ω be holomorphic coverings with h(0) = a

and k(0) = b. Because ∆ is not simply connected, the fiber h−1(a) is a discrete

subset of D and contains infinitely many points in addition to 0. Let 0 < r =min|z| : z ∈ h

−1(a), z 6= 0 < 1. The set h−1(a) ∩ z : |z| = r is finite, say

aj, 1 ≤ j ≤ m. Next, |z| : z ∈ k−1(b) is a discrete subset of [0, 1), so this set

contains finitely many values in the interval [0, r). Let s be the maximum valueof the finite set |z| : z ∈ k

−1(b)∩ [0, r). Suppose f : ∆ → Ω is any holomorphic

mapping with f(a) = b and f is not a covering. Let f be the unique lift of f h

relative to k that satisfies f(0) = 0. Then |f ′(0)| = f∆,Ω(a). Because f is not

a covering, f is not a rotation about the origin. From f h = k f we deduce


that f maps h−1(a) into k

−1(b). In particular, |f(a)| ≤ s = t|a|, where a = a1

and t = s/r < 1. Lemma 13.1 gives

|f ′(0)| ≤ (s/r) + |a|1 + (s/r)|a| =

(s/r) + r

1 + s= α < 1.

Since |f ′(0)| = f∆,Ω(a), this establishes (13.2). Inequality (13.3) follows imme-

diately from Corollary 11.3.

The pointwise result (13.2) is due to Minda [24] and was motivated by theAumann-Caratheodory Rigidity Theorem [4] which is the special case whenΩ = ∆ and a = b. The global result (13.3) is due to the authors [9]. The Aumann-Caratheodory Rigidity Theorem asserts there is a constant α = α(a, Ω) ∈ [0, 1)such that |f ′(a)| ≤ α for all holomorphic self-maps of Ω that fix a and are notconformal automorphisms. The exact value of the Aumann-Caratheodory rigid-ity constant for an annulus was determined in [23]. The following extension ofthe Aumann-Caratheodory Rigidity Theorem to a local result is due to the au-thors [9]. The corollary is given in Euclidean terms and asserts that holomorphicself-maps with a fixed point are locally strict Euclidean contractions if they arenot conformal automorphisms.

Corollary 13.3 (Aumann-Caratheodory Rigidity Theorem - Local Version).Suppose Ω is a hyperbolic region, a ∈ Ω and Ω is not simply connected. There

is a constant β = β(a, Ω) ∈ [0, 1) and a neighborhood N of a such that if f is a

holomorphic self-map of Ω that fixes a and is not a conformal automorphism of

Ω, then |f ′(z)| ≤ β for all z ∈ N .

Proof. From the theorem

|f ′(z)| ≤ λΩ(z)

λΩ(f(z))

α + tanh dΩ(a, z)

1 + α tanh dΩ(a, z).

Set M(r) = maxλΩ(z) : dΩ(a, z) ≤ r and m(r) = minλΩ(z) : dΩ(a, z) ≤ r.Since f(DΩ(a, r)) ⊆ DΩ(a, r), we have

|f ′(z)| ≤ M(r)

m(r)

α + tanh dΩ(a, z)

1 + α tanh dΩ(a, z).

The right-hand side of the preceding equality is independent of f and tends toα as r → 1. Therefore, for α < β < 1 there exists r > 0 such that

M(r)

m(r)

α + tanh dΩ(a, z)

1 + α tanh dΩ(a, z)≤ β

for dΩ(a, z) ≤ r. Then |f ′(a)| ≤ β holds in DΩ(a, r).

Our final topic is a rigidity theorem for holomorphic maps between annuli.The original results of this type are due to Huber [17]. Marden, Richards andRodin [21] presented an extensive generalization of Huber’s work to holomorphicself-maps of hyperbolic Riemann surfaces.


Definition 13.4. Suppose 1 < R,S ≤ +∞ and f : AR → AS is a continuousfunction. The degree of f is the integer deg f := I(f γ, 0), where γ(t) =exp(2πit).

The positively oriented unit circle γ generates the fundamental group of bothAR and AS. Therefore, for any continuous map f : AR → AS, f γ ≈ γ

n forthe unique integer n = deg f , where ≈ denotes free homotopy. Algebraically,f induces a homomorphism f∗ from π(AR, a) ∼= Z to π(AS, f(a)) ∼= Z and theimage of 1 is an integer n; here π(AR, a) denotes the fundamental group of AR

with base point a. The reader should verify that

deg(f g) = (deg f) (deg g).

Since the degree of the identity map is one, this implies that deg f = ±1 for anyhomeomorphism f . If f is holomorphic, then

deg f =1

2πi

∫

γ

f′(z)

f(z)dz.

Suppose f, g : AR → AS are continuous functions. Then deg f = deg g if andonly if f and g are homotopic maps of AR into AS.

Example 13.5. For any integer n the holomorphic self-map pn(z) = zn of C

∗

has degree n. Given annuli AR and AS with 1 < R,S < +∞, it is easy toconstruct a continuous map of AR into AS with degree n; for example, thefunction z 7→ (z/|z|)n has degree n. For R = S each conformal automorphismhas degree ±1. In fact, the rotations z 7→ e

iθz have degree 1 and the maps

z 7→ eiθ/z have degree −1. Constant self-maps of AR have degree 0. Can you find

a holomorphic self-map of AR with degree n 6= 0,±1? Surprisingly, the answeris negative! Holomorphic mappings of proper annuli are very rigid. The moduliof the annuli provide sharp bounds for the possible degrees of a holomorphicmapping of one annulus into another.

Theorem 13.6. If f : AR → AS is a holomorphic mapping, then

(13.4) | deg f | ≤ mod AS

mod AR

.

For n = deg f 6= 0 equality holds if and only if S = R|n| and f(z) = e

iθz

n for

some θ ∈ R.

Proof. Let γ(t) = exp(2πit) for t ∈ [0, 1]. If n = deg f , then I(f γ, 0) = n, soTheorem 12.2 gives

2π2|n|mod AS

≤ ℓS(f γ).

Since holomorphic functions are distance decreasing relative to the hyperbolicmetric,

ℓS(f γ) ≤ ℓR(γ) =2π2

mod AR

.

The preceding two inequalities imply (13.4). Suppose equality holds. Then f is acovering of AR onto AS that maps the unit circle onto itself. By post-composing


f with a rotation about the origin, we may assume that f fixes 1. Equalityin (13.4) implies S = R

|n|, where n = deg f . The covering f lifts relative to

pn(z) = zn to a holomorphic self-covering f of AR that fixes 1. Theorem 10.6

implies that f is the identity and so f(z) = zn.

Corollary 13.7 (Annulus Theorem). Suppose f is a holomorphic self-map of

AR. Then | deg f | ≤ 1 and equality holds if and only if f ∈ A(AR).

Proof. The inequality follows immediately from Theorem 13.6. If f ∈ A(AR),then | deg f | = 1 since this holds for any homeomorphism. It remains to showthat if | deg f | = 1, then f ∈ A(AR). Equality implies f maps the unit circle intoitself. By post-composing f with a rotation about the origin, we may assumef fixes 1. By Theorem 10.6 if a self-covering of a hyperbolic region has a fixedpoint, it is a conformal automorphism.

Note that if f is any holomorphic self-map of AR that is not a conformalautomorphism, then deg f = 0. A result analogous to Corollary 13.7 is notvalid for a punctured disk. For each integer n ≥ 0 the function z 7→ z

n is aholomoprhic self-map of D

∗ with degree n.

Exercises.

1. Show that Theorem 13.2 is false when ∆ is simply connected. Hint : Sup-pose ∆ = D and a = 0. For any number r ∈ [0, 1) show there exists aholomorphic function f : D → Ω that is not a covering and f

D,Ω(0) = r.2. Suppose f is a holomorphic self-map of C \ 0. Prove that deg f = 0 if

and only if f = exp g for some holomorphic function g defined on C \ 0.

14. Further reading

There are numerous topics involving the hyperbolic metric and geometric func-tion theory that are not discussed in these notes. The subject is too extensive toinclude even a reasonably complete bibliography. We mention selected books andpapers that the reader might find interesting. Anderson [3] gives an elementaryintroduction to hyperbolic geometry in two dimensions. Krantz [18] provides anelementary introduction to certain aspects of the hyperbolic metric in complexanalysis.

Ahlfors introduced the powerful method of ultrahyperbolic metrics [1]. Adiscussion of this method and several applications to geometric function theory,including a lower bound for the Bloch constant, can be found in [2]. Ahlfors’method can be used to estimate various types of Bloch constants, see [25], [26],[27]. In a long paper Heins [15] treates the general topic of conformal metrics onRiemann surfaces. He gives a detailed treatment of SK-metrics, a generalizationof ultrahyperbolic metrics. Roughly speaking, SK-metrics are to metrics withcurvature −1 as subharmonic functions are to harmonic functions.

The circle of ideas surrounding the theorems of Picard, Landau and Schottkyand Montel’s normality criterion all involve three omitted values. Theorems of


this type follow immediately from the existence of the hyperbolic metric on C∞punctured at three points. Interestingly, only a metric with curvature at most −1on a thrice punctured sphere is needed to establish these results. An elementaryconstruction of such a metric, based on earlier work of R. M. Robinson [31], isgiven in [30].

Hejhal [16] obtained a Caratheodory kernel-type of theorem for coverings ofthe unit disk onto hyperbolic regions. This result implies that the hyperbolicmetric depends continuously on the region.

The method of polarization was extended by Solynin to apply to the hyperbolicmetric, see [10]. It include earlier work of Weitsman [35] on symmetrization andMinda [29] on a reflection principle for the hyperbolic metric.

For the role of hyperbolic geometry in the study of discrete groups of Mobiustransformations, see [5]. This reference includes a brief treatment of hyperbolictrigonometry.

References

1. Ahlfors, L.V., An extension of Schwarz’s Lemma, Trans. Amer. Math. Soc., 43 (1938),259-264.

2. Ahlfors, L.V., Conformal invariants, McGraw-Hill, 1973.3. Anderson, J.W., Hyperbolic geometry, 2nd. ed., Springer Undergraduate Mathematics

Series, Springer, 2005.4. Aumann, G. and Caratheodory, C., Ein Satz uber die konforme Abbildung mehrfach

zusammenhangender Gebiete, Math. Ann., 109 (1934), 756-763.5. Beardon, A.F., The geometry of discrete groups, Graduate Texts in Mathematics 91,

Springer-Verlag, New York 1983.6. Beardon, A.F., The Schwarz-Pick Lemma for derivatives, Proc. Amer. Math. Soc., 125

(1997), 3255-3256.7. Beardon, A.F. and Carne, T.K., A strengthening of the Schwarz-Pick Inequality, Amer.

Math. Monthly, 99 (1992), 216-217.8. Beardon, A.F. and Minda, D., A multi-point Schwarz-Pick Lemma, J. d’Analyse Math.,

92 (2004), 81-104.9. Beardon, A.F. and Minda, D., Holomorphic self-maps and contractions, submitted.

10. Brock, F. and Solynin, A.Yu., An approach to symmetrization via polarization, Trans.Amer. Math. Soc., 352 (2000), 1759-1796.

11. Caratheodory, C., Theory of functions of a complex variable, Vol. II, Chelsea, 1960.12. Chen, Huaihui, On the Bloch constant, Approximation, complex analysis and potential

theory (Montreal, QC 2000), NATO Sci. Ser. II Math. Phys. Chem., 37, Kluwer Acad.Publ., Dordrecht, 2001.

13. Duren, P.L., Univalent functions, Springer-Verlag, New York 1983.14. Goluzin, G.M., Geometric theory of functions of a complex variable, Amer. Math. Soc.,

1969.15. Heins, M., On a class of conformal metrics, Nagoya Math. J., 21 (1962), 1-60.16. Hejhal, D., Universal covering maps for variable regions, Math. Z., 137 (1974), 7-20.

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Analyse Math., 18 (1967), 197-225.22. Massey, W.S., Algebraic topology: an introduction, Springer-Verlag, 1977.23. Minda, C.D., The Aumann-Caratheodory rigidity constant for doubly connected regions,

Kodai Math. J. 2 (1979), 420-426.24. Minda, C.D., The hyperbolic metric and coverings of Riemann surfaces, Pacific J. Math.

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Math. 13 (1983), 61-69.27. Minda, C.D., The hyperbolic metric and Bloch constants for spherically-convex regions,

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function theory, Complex Variables Theory Appl. 8 (1987), 129-144.30. Minda, D. and Schober, G., Another elementary approach to the theorems of Landau,

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A.F. Beardon E-mail: [email protected]: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,

Cambridge CB3 0WB, England

D. Minda E-mail: [email protected]: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio

45221-0025, USA


Isometries of relative metrics

Peter Hasto

Abstract. This survey contains a review of the basics of Mobius mappingsand some basic metrics used in relations with quasiconformal mappings, thejG metric and the quasihyperbolic metric. The second and third sections aredevoted to the study of the isometries of these two metrics.

Keywords. Relative metrics, isometries, distance ratio metric, Seittenranta’smetric.

2000 MSC. 30F45 (primary), 30C65 (secondary).

Contents

1. Overview 57

2. Mobius mappings 58

3. The jG metric 61

3.1. Isometries of j-type metrics 63

3.2. Other properties of j-type metrics 66

4. The quasihyperbolic metric 66

Additional notation 68

4.1. Isometries which are Mobius 68

4.2. Curvature of the quasihyperbolic metric 71

4.3. Isometries of the quasihyperbolic metric 73

References 76

1. Overview

These lecture notes consist of three parts: In the first part the basic theoryof Mobius mappings is reviewed. Particular emphasis will be given to concretecalculations within the context of a single mapping in Euclidean space. Althoughthis presentation is perhaps not the most elegant one possible, it has the advan-tage that it does a good job in preparing us for the isometry questions that comeup later. For a more detailed exposition of the basics of Mobius mappings seee.g. [2, 31].

Supported by the Academy of Finland.

58 Peter Hasto IWQCMA05

The latter two parts deal with the problem of characterizing isometries of twometrics which have turned out to be very important in the theory of quasicon-formal mappings, namely, the jG and the quasihyperbolic metric. Specifically, inthe second part we deal with the jG metric — this part is based on joint workwith Z. Ibragimov and H. Linden [18] in Computational Methods and FunctionTheory. The third part reproduces parts of my recent manuscript [15], whichdeals with isometries of the quasihyperbolic metric.

Characterizing isometries of a metric can in some sense be thought of as solvinga (system of) functional equation(s): we know that

df(G)(f(x), f(y)) = dG(x, y)

for all x, y ∈ G and we want to determine f . However, the fact that we have atour disposal a continuum of functional equations implies that the methods usedto approach this problem are somewhat different than those usually found whendealing with functional equations. Thus our methods will often be based onsome geometric considerations: we will employ geodesics (locally and globally),intrinsic curvature, as well as limiting behavior of the metric in infinitesimalregions more generally.

Many other properties of these and related metrics have also been studied. Areview of some of these results is presented in the chapter by H. Linden in thesenotes.

2. Mobius mappings

We denote by Rn = Rn∪∞ the one-point compactification of R

n, so its openballs are the open balls of R

n, complements of closed balls in Rn and half-spaces.

If D ⊂ Rn we denote by ∂D and D its boundary and closure, respectively, allwith respect to Rn. By B

n(x, r) and Sn−1(x, r) we denote the open ball centered

at x ∈ Rn with radius r > 0, and its boundary, respectively. For x ∈ D ( R

n wedenote δ(x) = d(x, ∂D) = min|x − z| : z ∈ ∂D. By [x, y], (x, y] we denote theclosed and half-open segment between x and y, respectively.

The (absolute) cross-ratio of four distinct points is defined by

|a, b, c, d| =|a − c| |b − d||a − b| |c − d| ,

with the understanding that |∞−x||∞−y| = 1 for all x, y ∈ R

n. A homeomorphism

f : Rn → Rn is a Mobius mapping if

|f(a), f(b), f(c), f(d)| = |a, b, c, d|for every quadruple of distinct points a, b, c, d ∈ Rn. A mapping of a subdomainof Rn is Mobius, if it is a restriction of a Mobius mapping defined on Rn.

Although the previous definition is very compact and brings out one importantaspect of Mobius mappings, it does not tell us what the behavior of a Mobiusmapping is in terms of geometry. However, it is not so difficult to get some

Isometries of relative metrics 59

results in this direction: Let us regard three points a, b, c as fixed and a fourthpoint x as variable. Then the cross-ratio equation reads

|a − c||a − b|

|b − x||c − x| =

|a′ − c′|

|a′ − b′||b′ − x

′||c′ − x′| ,

where a′ is the image of a under the Mobius mapping. We can rewrite this as

|b − x||c − x| = C

|b′ − x′|

|c′ − x′| ,

where C is a constant not depending on x. However, for fixed b, c and C > 0the set

x ∈ Rn :

|b − x||c − x| = C

is a sphere. Thus the previous equation implies that the Mobius mapping mapsspheres to spheres. The converse of this statement is also true, see [4].

It is also possible to take a more constructive approach to Mobius mappings.Let us first of all make the trivial observation that a mapping which preservesEuclidean distances is Mobius. Second, we note that mappings preserving ra-tios of Euclidean distances (so-called similarity mappings) are Mobius. Thesemappings are:

• translations;• reflections;• rotations; and• dilatations.

Are there any other Mobius mappings?

From the definition it is clear that the set of Mobius mappings is closed undercomposition (in fact, the set is a group under composition). Thus we may employa very useful trick in trying to identify any other Mobius mappings, namely, wenormalize by mappings that we already know are Mobius. This means that weconsider the mapping g = s1f s2, where f is our original Mobius mapping ands1 and s2 are similarities. Suppose first that f is such that f(∞) = ∞. Insertingd = ∞ in the definition implies that

|a − c||a − b| = |a, b, c,∞| = |f(a), f(b), f(c),∞| =

|f(a) − f(c)||f(a) − f(b)| ,

so f is a similarity. Otherwise there exists a finite point a such that f(a) = ∞.By an auxiliary similarity we may assume that a = 0 (i.e. we choose s2(x) = x+a

above). Similarly, f(∞) = b 6= ∞, and if we choose s1(x) = x − b, then g is aMobius mapping which swaps 0 and ∞. Using this in the equation gives

|b||c| = |∞, b, c, 0| = |0, g(b), g(c),∞| =

|g(c)||g(b)| .

From this we see that |x| |g(x)| is a constant. By another similarity we mayassume that this constant equals 1, so that |g(x)| = |x|−1 for every x ∈ R

n. To


get a grip of the non-radial action of g we use the equation inserting 0 and ∞ inother places:

|b − d||d| = |∞, b, 0, d| = |0, g(b),∞, g(d)| =

|g(b) − g(d)||g(b)|

. Using the previous formula for |g(b)| this gives

(2.1) |g(b) − g(d)| =|b − d||b| |d| ,

which is a central formula for calculating how a Mobius mapping affects distances.We can rewrite (2.1) as

|g(b) − g(d)|2|g(b)| |g(d)| =

|b − d|2|b| |d| .

Using the cosine formula

|b − d|2 = |b|2 + |d|2 − 2|b| |d| cos(b0d

),

where b0d stands for the angle between the vectors b−0 and d−0, and similarlyfor |g(b)−g(d)|2 we see that g preserves angles at the origin and lines through theorigin. Thus, up to additional normalization by a reflection and/or a rotation,we see that g(x) = x |x|−2.

A Mobius mapping which swaps ∞ and with a point of Rn and which maps

every line through this point to itself is called an inversion. Note that everyinversion is an involution, i.e. it is its own inverse. The point which is mapped to∞ is called the center of inversion. We have shown that every inversion equalsx 7→ x |x|−2, up to similarity. In particular, every Mobius mapping can be writtenas s i, where s is a similarity and i is an inversion or the identity.

Now that we have identified the Mobius mappings we can proceed to showthe following basic property: given two ordered triples of distinct points in Rn,(a, b, c) and (a′

, b′, c

′), there exists a Mobius map f with f(a) = a′, f(b) = b

′ andf(c) = c

′. It is clearly sufficient to show this claim in the case when a′, b

′ and c′

are the vertices of an equilateral triangle. Let us first find a point x such that

|a − c||a − b|

|b − x||c − x| =

|b − c||a − b|

|a − x||c − x| = 1.

The easiest way to see that such a point x exists is to use an inversion i withcenter a. Then the equations to satisfy become

|i(b) − z||i(c) − z| =

|i(b) − i(c)||i(c) − z| = 1.

We see that the first fraction describes a hyperplane which is the perpendicularbisector of the segment [i(b), i(c)] and the second fraction the sphere with centeri(c) and radius |i(b) − i(c)|. Since these objects clearly intersect, we can find a


suitable z, and then our x is given by i(z). Let ı be an inversion with center x.The choice of x implies that

|ı(a) − ı(c)||ı(a) − ı(b)| =

|ı(b) − ı(c)||ı(a) − ı(b)| = 1,

so(ı(a), ı(b), ı(c)

)are the vertices of an equilateral triangle. These can be

mapped to the given points (a′, b

′, c

′) by a similarity transform s, so our finalMobius map is then s ı.

3. The jG metric

This section reproduces parts of the article [18] on isometries of some relativemetrics. The term “relative metric” implies that the metric is evaluated ina proper subdomain of R

n relative to its boundary. More precisely, we wantthe metric to blow up towards the boundary of the domain, i.e., we want theboundary to be at infinity intrinsically.

Let D ( Rn be a domain containing the points x and y. The well-known

distance ratio metric is defined by

jD(x, y) = log

(1 +

|x − y|minδ(x), δ(y)

),

where δ(·) = dist( · , ∂D) denotes distance to the boundary. It was used, for in-stance, by Gehring and Osgood [11] to characterize uniform domains (namely, insuch domains the jD metric is quasiconvex). Note that this metric has sometimesbeen called simply “the relative metric”, and will be used in this meaning.

To see how these metrics fit into a larger framework we recall the concept ofan inner metric. Let d be a metric in D and γ be a path in D (i.e. a continuousmapping from an interval I to D). The length (or, more explicitly, d-length) ofγ is defined as

d(γ) = supk−1∑

i=1

d

(γ(ti), γ(ti+1)

),

where the supremum is taken over k and all increasing sequences (ti)ki=1

of pointsin I. Then the inner or intrinsic metric of d is defined by

d(x, y) = infγ

d(γ),

where the infimum is taken over all paths γ connecting x and y in D (notethat this need not be finite, unless D is rectifiably connected). It is clear that

d(x, y) ≤ d(x, y) and that d(γ) = d(γ) for any metric and path. The theory oflength-metrics, including in particular intrinsic metrics, is presented e.g. in [5, 6].

Suppose now that D ⊂ Rn and d is a metric in D. If

d(x) = limy→x

d(x, y)

|x − y|


exists for all x ∈ D and is continuous, then we can express the inner metric of d

by

d(x, y) = infγ

∫

γ

d(z) |dz|,

where |dz| represents integration with respect to d-arclength, and the infimum

is taken over rectifiable curves with end points x and y. In this case d is called aconformal metric. We easily see that the inner metric of jD is the quasihyperbolicmetric,

jD(x, y) = kD(x, y) = infγ

∫

γ

|dz|d(z, ∂D)

.

Length-metrics are interesting from a geometric point of view, but for gettingexplicit estimates they are often of little use. The role of point-distance functions,like the jD metric, is that they share features with their inner metrics, but aremuch more explicit.

In this paper we want to consider not only the jD metric, but all metricswhich resemble them in the very small and very large scale. The small scaleequivalence implies that the metrics have the same inner metrics, whereas thelarge scale equivalence allows us to get a hold of the boundary behavior and thusstart unraveling the isometry story.

Remark 3.1. Note that jD is really families of metrics, namely for every domainD we have one metric. We will continue to use this convention when talking aboutthis and other metrics in this paper.

Definition 3.2. We say that d is a j-type metric if the following three conditionshold on every domain D ( R

n:

1. dD is a metric on D.2. For each y ∈ D and for each sequence (xi) with jD(xi, y) → 0 we have

limi→∞

dD(xi, y)

jD(xi, y)= 1.

3. For each y ∈ D and for each sequence (xi) with jD(xi, y) → ∞ we have

limi→∞

(dD(xi, y) − jD(xi, y)

)= 0.

The fact that y can be any interior point in (2) means that being a j-typemetric is quite a strong condition; for instance, if d and f d are j-type metrics,then f = id (Corollary 3.12).

It seems to be quite difficult to construct other natural metrics of j-type. Themain purpose of our more abstract treatment is to highlight the features thatare crucial, which in turn indicates that these techniques might be relevant alsofor handling the isometries of the corresponding inner metrics.

In this part we characterize the isometries of j-type metrics. We start bycollecting some basic properties of these metrics. In Section 3.1 we solve the


isometry problem for j-type metrics using a boundary rigidity result, and inSection 3.2 we list some additional properties whose proofs can be found in [18].

The following result is more or less restatements of the definition. However,it directly implies that we may restrict our focus very much without losing anyisometries.

Proposition 3.3. If d is a j-type, then

limy→∂D\∞

(dD(x, y) − log

|x − y|δ(y)

)= 0 and lim

y→x

dD(x, y)

|x − y| =1

δ(x)

for every x ∈ D. The inner metric of a j-type metrics is the quasihyperbolic

metric kD. In particular, every isometry of a j-type metric is an isometry of kD.

Every isometry of kD is a conformal mapping [29, Theorem 2.6]. Hence weconclude:

Corollary 3.4. Every isometry of a j or δ-type metric is conformal. In partic-

ular, if n ≥ 3, then such an isometry is Mobius.

We plunge right into the main result of this section, a characterization of theisometries of j-type metrics. In Section 3.2 we derive some miscellaneous results,which give a clearer picture of j-type metrics.

3.1. Isometries of j-type metrics. The proof of the following theorem ispartly based on ideas from [17]. Incidentally, it is possible to give a much simplerproof for the particular case of the jD metric itself, since in this case we can cancelthe logarithm and the 1+ terms. Thus f is a jD isometry if and only if

|x − y|minδ(x), δ(y) =

|f(x) − f(y)|minδ′(f(x)), δ′(f(y)) ,

where, as usual, δ′ denotes the distance to the boundary in the image domain

f(D). The reader is challenged to find the very short argument which shows thatthis implies that the isometry is a similarity.

In the general case of j-type metrics we have less information about the metric,so we have to look at what happens at the boundary. In this case we cannevertheless prove the following theorem, whose proof is reproduced from [18].

Theorem 3.5. Let d be a j-type metric, D ( Rn and f : D → R

n be a d-

isometry. Then either

1. f is a similarity, or

2. D = Rn \ a and, up to similarity, f is an inversion in a sphere centered

at a.

Proof. Denote D′ = f(D) and δ

′(x) = d(x, ∂f(D)). Fix z ∈ ∂D \ ∞ and let(zi) be a sequence of points in D tending to z. We first assume that there existsa subsequence, which we also denote by (zi), such that

(f(zi)

)converges to some


point w1 ∈ Rn. Since d is a j-type metric we see, using Proposition 3.3 for the

third equality, that for every x ∈ D we have that

0 = limi→∞

(dD′(f(x), f(zi)) − dD(x, zi)

)

= limi→∞

(dD′(f(x), f(zi)) − log

1

δ′(f(zi))

)− lim

i→∞

(dD(x, zi) − log

1

δ(zi)

)+

+ limi→∞

logδ(zi)

δ′(f(zi))

= log|f(x) − w1||x − z| + lim

i→∞log

δ(zi)

δ′(f(zi)).

Taking exponentials gives

(3.6) limi→∞

δ′(f(zi))

δ(zi)=

|f(x) − w1||x − z| < ∞.

Suppose now that (zi) is a second sequence of points in D tending to z, butthat this time f(zi) → w2 ∈ Rn \ w1. Using x = zj for every j = 1, 2, . . . in(3.6) gives

limi→∞

δ′(f(zi))

δ(zi)=

|f(zj) − w1||zj − z| → ∞

as j → ∞, which is a contradiction. In other words, f(zi) → w1 for everysequence of points (zi) → z, so we may extend f continuously to D by definingf(z) = limi→∞ f(zi). Therefore we conclude from (3.6), since the left-hand sideof this equation does not depend on x, that

|f(x) − f(z)| = hf (z)|x − z|for some function hf : ∂D → (0,∞). This means that for z, w ∈ ∂D we have

hf (z)|w − z| = |f(w) − f(z)| = hf (w)|w − z|,so hf is in fact a constant. Therefore f acts as a similarity, say g, on the boundary.We then extend f to all of R

n by setting f(x) = g(x) outside the original domainof definition. Then it is clear that

(3.7) |f(x) − f(z)| = hf |x − z|for every point x ∈ R

n, i.e. the sphere Sn−1(z, r) is mapped to S

n−1(f(z), hfr).This clearly implies that the conformal mapping is Mobius, and a Mobius map-ping satisfying (3.7) is a similarity.

We still have one assumption to consider. In the beginning of the proof weassumed that we can find a boundary point z and a sequence (zi) of points inD tending to z such that f(zi) tends to a finite limit. So we suppose now thatno such sequence can be found, i.e. that for every sequence (zi) of points in D

tending to a boundary point z the sequence(f(zi)

)tends to ∞. As before we


conclude that

0 = limi→∞

(dD′(f(x), f(zi)) − dD(x, zi)

)

= limi→∞

(jD′(f(x), f(zi)) − jD(x, zi)

)

= limi→∞

log

( |f(x) − f(zi)|minδ′(f(zi)), δ′(f(x))

δ(zi)

|x − z|

).

So it follows that

limi→∞

|f(x) − f(zi)|δ(zi)

minδ′(f(zi)), δ′(f(x)) = |x − z|.

Since f(zi) → ∞, we see that we can replace |f(x)−f(zi)| by |f(zi)| in the aboveformula. Since the right-hand-side depends on x (which lies in an open set) wesee that the left-hand-side must do so, too, hence we have to choose the secondterm in the minimum. Taking this into account we have

gf (z) = limi→∞

|f(zi)| δ(zi) = |x − z| δ′(f(x)),

where gf : ∂D → (0,∞). Suppose that D has at least two finite boundary points,and let a, b ∈ ∂D be such that the open segment (a, b) is contained in D. Nowif we first consider x (in the previous equation) to be the mid-point x of (a, b),then we conclude that

gf (a) = |x − a| δ′(f(x)) = |x − b| δ′(f(x)) = gf (b).

But if we take some other point on the segment, then we get gf (a) 6= gf (b), acontradiction. So only the case when D has a single boundary point remains toconsider. Then we have

limi→∞

|f(zi)| |zi − a| = |x − a| |f(x) − b|

(for D = Rn \ a and D

′ = Rn \ b) and we directly see that x 7→ f(x) + b− a

is an inversion, which concludes the proof.

Corollary 3.8. Let d be a similarity invariant j-type metric and let D ( Rn.

Then f : D → Rn is a d-isometry if and only if

1. f is a similarity, or

2. D = Rn \ a and, up to similarity, f is the inversion in a sphere centered

at a.

Proof. The previous proposition established that every d-isometry is of the givenkind. If f is a similarity, then it is an isometry by assumption. So it remains(after normalization) to consider the case D = R

n \ 0. In this case we see that

similarity invariance implies that dD(x, y) depends only on max |x|

|y| ,|y||x|

and the

angle x0y. On the other hand, an inversion in a sphere about the origin swaps|x|/|y| and |y|/|x| and leaves x0y invariant, so we see that it is an isometry.


3.2. Other properties of j-type metrics. We said before that every j-typemetric has an upper bound in terms of the quasihyperbolic metric. Surprisingly,it is also possible to get a universal lower bound by a metric, the so-called half-

apollonian metric [19]. For a domain D ( Rn this metric is defined by

ηD(x, y) = supz∈∂D

∣∣∣∣ log|x − z||y − z|

∣∣∣∣.

The metric ηD is similarity invariant, and every Mobius mapping is bilipschitz.

The proofs of the following results can be found in [18].

Proposition 3.9. For every j-type metric d and every D ( Rn we have dD ≥ ηD.

Using the previous proposition and the quasihyperbolic upper bound we cansqueeze in j-type metrics to get the exact value on some subset of the domain:

Corollary 3.10. Let w ∈ D and z ∈ ∂D ∩ Sn−1(w, δ(w)). Then dD(x, y) =

jD(x, y) for every x, y ∈ [w, z).

The following is easily checked by a direct computation using the definition ofthe j-metric, but also follows from Corollary 3.10 and the fact that line segmentsare also geodesic rays for the ηD-metric ([19, Example 3.4]).

Corollary 3.11. Let w ∈ D and z ∈ ∂D ∩ Sn−1(w, δ(w)). Then [w, z) is a

geodesic ray for the j-type metric d, i.e. for every x, ξ, y ∈ [w, z) in this order we

have

dD(x, y) = dD(x, ξ) + dD(ξ, y).

In general, if we have a metric d and a subadditive function f : [0,∞) → [0,∞)for which f(x) = 0 if and only if x = 0, then f d is also a metric. It turns outthat the conditions for begin a j-type metric are so rigid, that this transformationis never possible in this context:

Corollary 3.12. Let d be a j-type metric and fD : [0,∞) → [0,∞) be a family

of arbitrary functions. If f d is a j-type metric, then fD = id for all relevant

D.

4. The quasihyperbolic metric

The remainder of this article is reproduced with minor modifications from[15].

Let D ( R2 be an open set and denote δ(x) = d(x, ∂D), the distance to the

boundary. The quasihyperbolic metric in D is the conformal metric with thedensity δ(x)−1, in other words, the metric is given by

kD(x, y) = infγ

∫

γ

ds(z)

δ(z),

where the infimum is taken over paths γ connecting x and y in D and ds repre-sents integration with respect to arc-length.


The quasihyperbolic metric was first introduced in the seventies, and sincethen it has found innumerable applications, especially in the theory of quasicon-formal mappings, see, e.g. [11, 12, 22, 28, 29]; new connections are still beingmade, for instance P. Jones and S. Smirnov [24] recently gave a criterion for re-movability of a set in the domain of definition of a Sobolev space in terms of theintegrability of the quasihyperbolic metric, see also [25], and Z. Balogh and S.Buckley [1] used the metric in a geometric characterization of Gromov hyperbolicspaces.

Despite the prominence of the quasihyperbolic metric, there have been almostno investigations of its geometry. Three exceptions are the papers by G. Martin[28] and Martin and B. Osgood [29], the second of which was the main motivationfor the approach presented in this paper, and the thesis by H. Linden [27]. Part ofthe reason for this lack of geometrical investigations is probably that the densityof the quasihyperbolic metric is not differentiable in the entire domain, whichplaces the metric outside the standard framework of Riemanian metrics.

At least two modifications of the quasihyperbolic metric have been proposedwhich do not suffer from this problem. J. Ferrand [10] suggested replacing thedensity δ

−1 by

σD(x) = supa,b∈∂D

|a − b||a − x| |b − x| .

Note that δ(x)−1 ≤ σD(x) ≤ 2δ(x)−1, so the Ferrand metric and the quasihyper-bolic metric are bilipschitz equivalent. Moreover, the Ferrand metric is Mobiusinvariant, whereas the quasihyperbolic metric is only Mobius quasi-invariant. Asecond variant was proposed more recently by R. Kulkarni and U. Pinkall [26],see also [23]. The K–P metric is defined by the density

µD(x) = inf 2r

(r − |x − z|)2: x ∈ B(z, r) ⊂ D

.

Equivalently, the infimum is taken over the hyperbolic densities of x in ballscontained in D. This density satisfies the same estimate as Ferrand’s density,i.e. δ(x)−1 ≤ µD(x) ≤ 2δ(x)−1, and the K–P metric is also Mobius invariant.Although the Ferrand and K–P metrics are in some sense better behaved thanthe quasihyperbolic metric, they suffer from the short-coming that it is verydifficult to get a grip even of the density, even in simple domains.

Despite this, D. Herron, Z. Ibragimov and D. Minda [21] recently managed tosolve the isometry problem of the K–P metric in most cases. By the isometryproblem of the metric d we mean characterizing mappings f : D → R

2 with

dD(x, y) = df(D)(f(x), f(y))

for all x, y ∈ D. Notice that in some sense we are here dealing with two dif-ferent metrics, due to the dependence on the domain. Hence the usual way ofapproaching the isometry problem is by looking at some intrinsic features of themetric which are then preserved under the isometry. Since irregularities (e.g.cusps) in the domain often lead to more distinctive features, this implies thatthe problem is often easier for more complicated domains.


The work by Herron, Ibragimov and Minda [21] bears out this heuristic – theywere able to show that all isometries of the K–P metric are Mobius mappingsexcept in simply and doubly connected domains. Their proof is based on studyingthe curvature of the metric. For the quasihyperbolic metric, formulae for thecurvature were worked out already in [29] (see Section 4.2, below), and wereused in that paper to prove that all the isometries of the disc are similaritymappings. These will be our main tool in this paper. The other source of theideas used below are the papers [16, 17, 18, 19] on isometries of some othersimilarity and Mobius invariant metrics.

There are three steps in characterizing quasihyperbolic isometries:

1. show that they are conformal;2. show that they are Mobius; and3. show that they are similarities.

The first step has been carried out by Martin and Osgood [29, Theorem 2.6] forcompletely arbitrary domains, so there is no more work to do there. In Section 4.3we will use the results from [29] on the curvature of the quasihyperbolic metric,and some new ideas to prove that the conformal isometries are Mobius (secondstep). For this we need to assume that the boundary of the domain is at leastC

3-smooth. In Section 4.1 we will work on the third step – we show that Mobiusisometries are similarities provided the boundary is C

1. In Section 4.2 we studythe Gaussian curvature of the quasihyperbolic metric, and the gradient of thecurvature.

Additional notation. We employ some additional conventions in this section:We tacitly identify R

2 with C, and speak about real and imaginary axes, etc.We will often work with a mapping f : D → R

2. In such cases we will use aprime to denote quantities on the image side, e.g. x

′ = f(x), D′ = f(D) and

δ′(x) = d(x, ∂D

′), and so on.

4.1. Isometries which are Mobius. Let D be a domain and ζ ∈ ∂D. We saythat ζ is circularly accessible, if there exists a disc B ⊂ D such that ζ ∈ ∂B.

Lemma 4.1. Let D ( R2 be a Jordan domain with circularly accessible bound-

ary, and let f : D → R2 be a quasihyperbolic isometry which is also Mobius.

Then, up to composition by similarity mappings, f is the identity or the inver-

sion in a circle centered at a boundary point.

Proof. Assume that f is not a similarity. Since f is a Mobius map, it is aninversion, up to similarities, which are always isometries of the quasihyperbolicmetric. Thus it suffices to consider the case when f is an inversion in a unitsphere. Let us denote the center of this sphere by w.

Suppose first that w 6∈ D and let ζ ∈ ∂D be the closest boundary point to w.For simplicity we normalize the situation so that ζ lies on the positive real axisand w = 0. Since ζ is circularly accessible, we find a disc B(z, r) ⊂ D whichcontains ζ in its closure. Since ζ is the closest boundary point to w, we see that


z has to lie on the positive real axis, as well. Let x and y be points satisfying

ζ < x < y ≤ ζ(ζ+2r)

ζ+r. The right-hand inequality ensures that ζ is the closest

boundary point to [x, y], and that ζ′ is the closest boundary point to [x′

, y′].

Thus we find that

kD(x, y) = log|x − ζ||y − ζ| and kD′(x′

, y′) = log

|x′ − ζ′|

|y′ − ζ ′| .

Since f is the inversion in the unit sphere, we have

|x′ − ζ′| =

|x − ζ||x| |ζ| ,

and similarly for y. Then the equation exp kD(x, y) = exp kD′(x′, y

′) gives us

|x − ζ||y − ζ| =

|x − ζ||x| |ζ|

|y| |ζ||y − ζ| ,

i.e. |x| = |y|. This contradiction shows that w ∈ D. Since f maps D into R2, it

is clear that w 6∈ D, so it follows that w is a boundary point.

We call D a Ck domain, if ∂D is locally the graph of a C

k function. Note thatif D is a C

1 domain, then certainly every boundary point is circularly accessible.

Proposition 4.2. Let D ( R2 be a C

1 domain, and let f : D → R2 be a quasi-

hyperbolic isometry which is also Mobius. If D is not a half-plane, then f is a

similarity.

Proof. We assume that f is not a similarity map. By the previous lemma wesee that there is no loss of generality in considering only the case when f isthe inversion centered at a boundary point. For simplicity of exposition, wenormalize so that the origin is this center.

Let ζ be a boundary point of D distinct from 0 and let u be the inwardpointing unit normal at ζ. For all sufficiently small t > 0, the point xt = ζ + tu

lies in D and its closest boundary point is ζ. For such s < t, we have

kD(xt, xs) = logt

s.

To estimate the distance of the image points, we use the inequality

jD′(x′, y

′) = log

(1 +

|x′ − y′|

minδ′(x′), δ′(y′)

)≤ kD′(x′

, y′),

which is always valid (since kD′ is the inner metric of jD′ , e.g. [12, Lemma 2.1]).We also need the formula

|x′ − y′| =

|x − y||x| |y|


for the length distortion of an inversion. Using these facts and the estimateδ′(x′) ≤ |x′ − ζ

′|, we derive the inequality

kD′(x′, y

′) ≥ log(1 +

|x′ − y′|

minδ′(x′), δ′(y′))

≥ log(1 +

|x − y|/(|x| |y|)min|x′ − ζ ′|, |y′ − ζ ′|

)

= log(1 +

|x − y| |ζ||x| |y| min|x − ζ|/|x|, |y − ζ|/|y|

)

= log(1 +

|x − y| |ζ|min|y| |x − ζ|, |x| |y − ζ|

).

Applying this inequality to the points xt and xs as defined before, we have

kD′(x′t, x

′s) ≥ log

(1 +

(t − s) |ζ|mint |xs|, s |xt|

).

Let us choose t = 2s. Since |x2s| and |xs| both tend to |ζ| as s → 0, we see thatthe second term in the minimum is smaller. Since the inversion is supposed to bean isometry, we can use the formula for kD(xt, xs) from before with the previousinequality to conclude that

log2s

s≥ log

(1 +

(2s − s) |ζ|s |x2s|

).

Taking the exponential function gives |x2s| ≥ |ζ|. Since xs = ζ + su, this impliesthat 〈ζ − 0, u〉 ≥ 0 as s → 0, where 〈, 〉 denotes the scalar product.

Applying the same argument, but starting with points on the image side, weconclude that the opposite inequality is also valid. (There is actually a slightasymmetry here: the domain D

′ need not have circularly accessible boundaryat the origin. However, it is clear that this does not affect the argument sofar.) Thus it follows that 〈ζ − 0, u〉 = 0 for all boundary points. But since theboundary is assumed to be C

1, this implies that the domain is a half-plane.

From [29, Theorem 2.8] we know that if f : D → R2 is a quasihyperbolic isom-

etry, then f is conformal in D. In dimensions three and higher every conformalmapping is Mobius. It is easy to see that the proofs in this section work also inthe higher dimensional case. Therefore, we have proved the following result:

Corollary 4.3. Let D be a C1 domains in R

n, n ≥ 3, which is not a half-space.

Then every quasihyperbolic isometry is a similarity mapping.

Example 4.4. Note that if we do not assume C1 boundary, then there are some

further domains with non-trivial isometries: the punctured planes R2 \ a and

sector domains (i.e. domains whose boundary consists of two rays). In bothcases inversions centered at the distinguished boundary point (a or the vertexof the sector). The previous proposition strongly suggests that these are all theexamples.


4.2. Curvature of the quasihyperbolic metric. Let D be a domain in R2.

We call a disc B ⊂ D maximal, if it is not contained in any other disc containedin D. The set consisting of the centers of all maximal discs in D is called themedial axis of D and denoted by MA(D). The medial axis and differentiabilityproperties of the distance-to-the-boundary function have been studied e.g. in[7, 8, 9].

In a general domain the Gaussian curvature of the quasihyperbolic metric isnot defined, since the distance-to-the-boundary function is not C

2. M. Heins[20] considered this situation for a quite general class of metric, and defined thenotions of upper and lower curvature. Martin and Osgood worked with thesecurvatures in the context of the quasihyperbolic metric, see [29, Section 3] fordetails. However, if our domain is sufficiently regular (say C

2), and we areconsidering points not on the medial axis, then the upper and lower curvatureagree, and define the curvature. In this case the curvature of kD is given by

KD(z) = −δ(z)2 log δ(z),

[20, (1.3)] or [29, (3.1)]. On the medial axis this formula does not make sense,but the upper and lower curvatures still agree, and both equal −∞, by [29,Corollary 3.12].

The next lemma is a specialization of Lemma 3.5, [29] to the case there theupper and lower curvatures agree.

Lemma 4.5 (Lemma 3.5, [29]). Let G and G be C2 domains such that B(z, r) ⊂

G ∩ G and ζ ∈ (∂G) ∩ (∂G) ∩ (∂B(z, r)). If there is a neighborhood U of ζ such

that G ∩ U ⊂ G ∩ U and d(z, ∂G \ U) > d(z, ∂G), then KG(z) ≤ KG(z).

Using this lemma we can derive the following very plausible statement, whichsays that the Gaussian curvature of the quasihyperbolic metric depends only onthe curvature of the boundary at the closest boundary point. We sill need somemore notation.

Let B be a disc with ζ ∈ (∂B)∩ (∂D). Then we call B the osculating disc atζ if ∂B and ∂D have second order contact at ζ. Let D be at least a C

2 domain.Then there exists an osculating disc at every boundary point ζ. If this disc hasradius r, then we define Rζ to be r if the disc lies in the direction of the interior ofD, and −r otherwise. Note that the function ζ 7→ 1/Rζ is C

k−2 in a Ck domain,

k ≥ 2.

Proposition 4.6. Let D ( R2 be a C

2 domain and z ∈ D \MA(D) have closest

boundary point ζ ∈ ∂D. Then

KD(z) = − Rζ

Rζ − δ(z)= − 1

1 − δ(z)/Rζ

.

If z lies on the medial axis, then KD(z) = −∞.

Proof. The medial axis consists of points equidistant to two or more nearestboundary points, and of centers of osculating circles. For the former, the claim


that KD(z) = −∞ follows from [29, Corollary 3.12]. So we assume that z has aunique nearest boundary point, ζ.

We suppose further that Rζ > 0, the other case begin similar. Let B(w,Rζ)be the osculating disc at ζ. We define

Bt = B(w + w−ζ

Rζ

t, Rζ + t),

and note that ∂Bt contains ζ for all t > −Rζ . We have the formula

KB(0,r)(x) = − r

|x| = − r

r − d(x, ∂B(0, r))

for the curvature of the quasihyperbolic metric in a ball [29, Lemma 3.7], so wecan calculate KBt

(z) explicitly.

Using the previous lemma with G = D and G = Bt for t > 0 gives KD(z) ≤KBt

(z). If z is the center of B0, then right-hand-side of this inequality tendsto −∞ as t → 0, which completes the proof of the claim regarding the medialaxis. So we assume that z is not the center of B0, and then we can apply theLemma 4.5 with G = Bt for t < 0 (sufficiently close to 0) and G = D to getKBt

(z) ≤ KD(z). Thus we have

KB−t(z) ≤ KD(z) ≤ KBt

(z)

for small t > 0. Since KBtis continuous in t, we get KD(z) = KB0

(z) as we lett → 0. The proof is completed by applying the aforementioned formula for thecurvature to the ball B0 = B(w,Rζ).

Let f : D → R2 be a C

1 mapping. By ∇f we denote the gradient of f ,i.e. the vector (∂1f, ∂2f), and by ∇f(z) we denote δ(z)∇f(z). The reason formultiplying by δ(x) is that

δ(y) = limx→y

|x − y|kD(x, y)

,

so that the ∇ operator is more natural in the setting where the quasihyperbolicbut not the Euclidean distance is preserved (see (4.9), below).

We next present an explicit formula for ∇KD. For this need a mapping whichassociates to every point in D \ MA(D) its closest boundary point. We call thismapping ζ = ζ(z).

Lemma 4.7. Let D ( R2 be a C

3 domain. Then

∇KD(z) = (KD(z) + 1)[KD(z)∇δ(z) − (KD(z) + 1)∇Rζ(z)

]

for every z off the medial axis, where all differentiation is with respect to the

variable z.

Proof. We use the formula from Proposition 4.6. Thus

∇KD(z) = −∇ 1

1 − δ(z)/Rζ

= KD(z)2∇δ(z)

Rζ

=KD(z)2

R2

ζ

(Rζ∇δ(z) − δ(z)∇Rζ),


where we understand ζ as a function of z. Note that Rζ and δ are C1, since D

is C3 and we are not on the medial axis. From Proposition 4.6 we also get

δ(z)

Rζ

=KD(z) + 1

KD(z).

Thus we continue the equation by

∇KD(z) = KD(z)2δ(z)

Rζ

(∇δ(z) − δ(z)

Rζ

∇Rζ

)

= (KD(z) + 1)(KD(z)∇δ(z) − (KD(z) + 1)∇Rζ

).

We next show that |∇K| is an intrinsic quantity of the quasihyperbolic metric.

Lemma 4.8. Let D be a C3 domain. If f : D → R

2 is a quasihyperbolic isometry,

then |∇KD(z)| = |∇Kf(D)(f(z))| for every z ∈ D.

Proof. We know that f is conformal. For a unit vector u we find that

⟨∇KD(z), u

⟩= lim

ε→0

KD(z + εu) −KD(z)

kD(z + εu, z)

= limε→0

Kf(D)(f(z + εu)) −Kf(D)(f(z))

kf(D)(f(z + εu), f(z)).

(4.9)

Next we note that f(z +εu) = f(z)+εf′(z)u+O(ε2). Here f

′(z)u is understood

as complex multiplication. Let us define another unit vector u = f ′(z)

|f ′(z)|u. Then

we continue the previous equation by

⟨∇KD(z), u

⟩= lim

ε→0

Kf(D)(f(z) + εf′(z)u) −Kf(D)(f(z))

kf(D)(f(z) + εf ′(z)u, f(z))

= limε→0

ε|f ′(z)|〈∇Kf(D)(f(z)), u〉ε|f ′(z)|δ′(f(z))−1

=⟨∇Kf(D)(f(z)), u

⟩.

Since u was an arbitrary unit vector, we see that |∇KD(z)| = |∇Kf(D)(f(z))|.

4.3. Isometries of the quasihyperbolic metric. We know that similaritiesare always quasihyperbolic isometries, and we want to show that in most casesthese are the only ones. In view of the results in Section 4.1, it suffices for us toshow that a quasihyperbolic isometry is a Mobius mapping, so this will be whatwe aim at in the proofs of this section.

A curve γ in D is a (quasihyperbolic) geodesic if

kD(x, y) = kD(x, z) + kD(z, y)

for all x, z, y ∈ γ in this order. It is clear from this definition that geodesics arepreserved by isometries. A geodesic ray is a geodesic which is isometric to R

+.For every z ∈ D we easily find one geodesic ray, namely [z, ζ(z)), which alsohappens to be a Euclidean line segment. The idea is to show that this geodesic


is somehow special (from a quasihyperbolic point-of-view), so that it would mapto a geodesic ray of the same kind.

Lemma 4.10. Let D ( R2 be a C

2 domain with a boundary point ξ such that

1/Rξ = 0. Then every isometry f : D → R2 of the quasihyperbolic metric is

Mobius.

Proof. Let B ⊂ D be a non-maximal disc whose boundary contains ξ and letz denote the center of B. By Proposition 4.6 we find that KD ≡ −1 on thesegment γ = [z, ξ). Thus Kf(D) ≡ −1 on γ

′, so 1/R′ζ′(z′) = 0 for every point z

′

on this curve. We consider two cases: either ζ′(z′) is just a single point for all

z′ ∈ γ

′, or it sweeps out a non-degenerate subcurve of the boundary ∂D′ as z

′

varies over γ′. (There is no third possibility, since ζ

′ is a continuous function onγ′.) In the single-point case we see that γ

′ has to be a line segment, since theboundary does not have corners. In this case we find that

kD(x, y) =∣∣ log |x−ξ|

|y−ξ|

∣∣ and kD′(x′, y

′) =∣∣ log |x′−ξ′|

|y′−ξ′|

∣∣,where ξ

′ is the closest boundary point to the every point on γ′. But this easily

implies that f is Mobius on γ. Since f is conformal it follows by uniqueness ofanalytic extension that f is a Mobius mapping on all of D.

So we consider the second case, that ζ′(z′) sweeps out a non-degenerate sub-

curve of the boundary ∂D′. Since the curvature of the boundary at all these

points is zero, it follows that the piece of the boundary is a line segment, L′.

Let U′ ⊂ D

′ be an open set such that (∂U′) ∩ (∂D

′) = L′ and the nearest

boundary point of every point in U′ lies in L

′. Then the geometry of the quasi-hyperbolic metric in U is the same as in a half-plane, in particular KD′ ≡ −1on U

′. Then KD ≡ −1 on U = f−1(U ′), so it follows that (∂U) ∩ (∂D) = L, for

some line segment L. So it follows that f |U is the restriction of a quasihyper-bolic isometry of the half-plane. But these are only the Mobius mappings. Thenwe again conclude from the uniqueness of analytic extension that f is a Mobiusmapping on all of D.

Let us call a domain strictly concave, if its complement is strictly convex.

Corollary 4.11. Let D ( R2 be a C

2 domain which is not a half-plane, strictly

convex or strictly concave. Then every quasihyperbolic isometry is a similarity

mapping.

Proof. Suppose that 1/Rζ 6= 0 for all boundary points. Since 1/Rζ is continuousby assumption, this implies that it is either everywhere positive, or everywherenegative. In these cases we have a strictly convex and strictly concave domain,respectively, which was ruled out by assumption. So we find some point at which1/Rζ = 0. Then it follows from Lemma 4.10 that the isometry is Mobius andfrom Lemma 4.2 that it is a similarity.

So we are left with only two types of domains that we cannot handle: strictlyconvex and strictly concave ones. As usual when working with isometries, the


nicest domains turn out to be the most difficult. Unfortunately, we need toassume more regularity of the boundary in order to take care of these cases.

Theorem 4.12. Let D ( R2 be a C

3 domain, which is not a half-plane. Then

every isometry f : D → R2 of the quasihyperbolic metric is a similarity mapping.

Proof. In view of Corollary 4.11, we may restrict ourselves to the case whenKD(z) 6= −1 for all z ∈ D. Let z ∈ D \ MA(D) and ζ be its nearest boundarypoint. We note that ∇δ(z) and ∇Rζ are perpendicular – first of all, ∇δ(z) isparallel to z − ζ; second, Rζ is a constant in the direction of z − ζ, since ζ is theclosest boundary point to all points on this line (near z).

If D is bounded, then it is clear that Rζ has a critical point. If D is unbounded,then we note that 1/Rζ cannot have any other limit than 0 at ∞ (although alimit need not exist, of course). Thus we see that Rζ has a critical point in theunbounded case as well. Let ζ be a critical point of ξ 7→ Rξ and fix a point z ∈ D

with KD(z) 6= −∞ whose nearest boundary point is ζ. Of course, ∇Rζ = 0 atthe critical point ζ. Then it follows from Lemma 4.7 that

∇KD(z) = (KD(z) + 1)KD(z)∇δ(z).

Since the curvature is intrinsic to the metric, we have KD′(z′) = KD(z). Also,|∇KD′(z′)| = |∇KD(z)| by Lemma 4.8, so we have∣∣(KD(z) + 1)KD(z)∇δ(z)

∣∣ =∣∣(KD(z) + 1)

[KD(z)∇δ

′(z′)− (KD(z) + 1)∇R′ζ′(z′)

]∣∣.We know that KD(z) 6= −1 and that ∇δ

′(z′) and ∇R′ζ′(z′) are orthogonal. Thus

the previous equation simplifies to(KD(z)|∇δ(z)|

)2

=(KD(z)|∇δ

′(z′)|)2

+((KD(z) + 1)

∣∣∇R′ζ′(z′)

∣∣)2

.

Since |∇δ| = 1 off the medial axis for every domain, this equation implies that∇Rζ′ = 0.

So for our point z, ∇KD(z) and ∇KD′(z′) point to the nearest boundarypoint of z and z

′, respectively. Let γ = [z, ζ). Note that γ is a geodesic of thequasihyperbolic metric. Also, ∇KD(z) and γ are parallel at z. Now γ is mappedto some geodesic ray γ

′, and since f is a conformal mapping, γ′ is parallel to

∇KD′(z′) at z′. But [z′, ζ ′) is a geodesic parallel to ∇KD′(z′) at z

′, and sincegeodesics are unique (when the density is C

2, i.e. except possibly on the medialaxis) we see that γ

′ = [z′, ζ ′).

So we have shown that f([z, ζ)) = [z′, ζ ′). Moreover, we have

kD(x, y) =∣∣∣ log |x−ζ|

|y−ζ|

∣∣∣ and kD′(x′, y

′) =∣∣∣ log |x′−ζ′|

|y′−ζ′|

∣∣∣

for x, y ∈ [z, ζ). Thus we see that f is just a similarity on [z, ζ). But f is aconformal map, so this implies that f is a similarity in all of D.

Acknowledgment. I would like to thank Zair Ibragmov for several discussionsabout the isometries of this and related metrics and Swadesh Sahoo for somecomments on this manuscript.


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1137–1142.[15] : Isometries of the quasihyperbolic metric, submitted.[16] P. Hasto and Z. Ibragimov: Apollonian isometries of planar domains are Mobius mappings,

J. Geom. Anal. 15 (2005), no. 2, 229–237.[17] : Apollonian isometries of regular domains are Mobius mappings, Ann. Acad. Sci.

Fenn. Math., to appear.[18] P. Hasto, Z. Ibragimov and H. Linden: Isometries of relative metrics, Comput. Methods

Funct. Theory 6 (2006), no. 1, 15–28.[19] P. Hasto and H. Linden: Isometries of the half-apollonian metric, Complex Var. Theory

Appl. 49 (2004), 405–415.[20] M. Heins: On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60.[21] David Herron, Zair Ibragimov and David Minda: Geometry of the K-P metric, preprint

(2005).[22] D. Herron and P. Koskela: Conformal capacity and the quasihyperbolic metric, Indiana

Univ. Math. J. 45 (1996), no. 2, 333–359.[23] D. Herron, W. Ma and D. Minda: A Mobius invariant metric for regions on the Riemann

sphere, pp. 101–118 in Future Trends in Geometric Function Theory (RNC Workshop,Jyvaskyla 2003; D. Herron (ed.)), Rep. Univ. Jyvaskyla Dept. Math. Stat. 92 (2003).


[24] P. Jones and S. Smirnov: Removability theorems for Sobolev functions and QC maps,Ark. Mat. 38 (2000), no. 2, 263–279.

[25] P. Koskela and T. Nieminen: Quasiconformal removability and the quasihyperbolic metric,Indiana Univ. Math. J. 54 (2005), no. 1, 143–151.

[26] R. Kulkarni and U. Pinkall: A canonical metric for Mobius structures and its applications,Math. Z. 216 (1994), 89–129.

[27] H. Linden: Quasihyperbolic Geodesics and Uniformity in Elementary Domains, Ph.D.Thesis, University of Helsinki, 2005.

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(1999), 511–533.[31] M. Vuorinen: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math.,

Vol. 1319, Springer, Berlin, 1988.

Peter Hasto E-mail: [email protected]: Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu,

Finland


Uniform Spaces and Gromov Hyperbolicity

David A Herron

Abstract. This brief outline contains mostly definitions, background infor-mation, and statements of theorems. Along with the title topics, we alsodiscuss the uniformization volume growth problem as well as certain capacityand slice condition characterizations of uniformity.

Keywords. uniform spaces, Gromov hyperbolicity, quasihyperbolic metric,volume growth, Ahlfors regular spaces, Loewner spaces, slice conditions.

2000 MSC. Primary: 30C65; Secondary: 53C23, 30F45.

Contents

1. Introduction 80

2. Metric Space Background 84

2.A. General Information 85

2.B. Abstract Domains 86

2.C. Maps and Gauges 87

2.D. Length and Geodesics 88

2.E. Connectivity Conditions 89

2.F. Doubling and Dimensions 90

2.G. Quasiconformal Deformations 92

2.H. Quasihyperbolic Distance and Geodesics 96

2.I. Modulus and Capacity 97

2.J. Ahlfors Regular and Loewner Spaces 99

2.K. Slice Conditions 101

3. Uniform Spaces 102

3.A. Euclidean Setting 102

3.B. Measure Metric Space Setting 102

3.C. Basic Information 103

4. Gromov Hyperbolicity 103

4.A. Thin Triangles Definition 104

Version October 19, 2006.The author was supported by the NSF and the Charles Phelps Taft Research Center.

80 D.A.Herron IWQCMA05

4.B. Gromov Boundary 104

4.C. Connection with Uniform Spaces 105

5. Uniformization 105

5.A. Uniformization Problem 105

5.B. BHK Uniformization 105

5.C. Bounded Geometry and its Consequences 107

5.D. Lifts and Metric Doubling Measures 109

5.E. Volume Growth Problem 110

6. Characterizations of Uniform Spaces 111

6.A. Metric Characterizations 111

6.B. Gromov Boundary Characterizations 111

6.C. Characterizations using QC Maps 112

6.D. Capacity Conditions 112

6.E. LLC and Slice Conditions 113

References 113

1. Introduction

This survey is meant to supplement the talks I presented at the International

Workshop on Quasiconformal Mappings and their Applications and at the Interna-

tional Conference on Geometric Function Theory, Special Functions and their Appli-

cations. Primarily, I provide here basic background material including definitions,terminology, and fundamental facts. I also list a few references, many of whichthemselves contain additional references to this material. I have made no at-tempt to render a complete list of references and apologize to all those whosework I have neglected to mention. The reader is absolutely encouraged to consultthe many works referred to by the authors I do mention.

The goal of these notes is to provide the reader with a foundation enablingthem to understand the meaning and relevance of the recent work [BHK01],[BHR01], [BKR98] of Bonk, Heinonen, Koskela and Rohde along with [Her04]and [Her06]. I am delighted to thank Mario Bonk, Juha Heinonen and PekkaKoskela for numerous helpful discussions and hours of blackboard sessions re-garding these topics.

By now Euclidean uniform spaces (domains in Euclidean space in which pointscan be joined by short twisted double cone arcs) are well recognized as being the‘nice’ spaces for quasiconformal function theory as well as many other areas ofanalysis (e.g., potential theory); see [Geh87], [Vai88] for Euclidean space and[Gre01], [CGN00], [CT95] for the Carnot-Caratheodory setting. In [BHK01]Bonk, Heinonen, and Koskela develop a uniformization theory which provides a

Uniformity and Hyperbolicity 81

two way correspondence between uniform spaces and Gromov hyperbolic spaces.In particular, they prove the following fundamental result; see [BHK01, Theorem1.1].

There is a one-to-one (conformal) correspondence between quasiisom-

etry classes of proper geodesic roughly starlike Gromov hyperbolic spaces

and quasisimilarity classes of bounded locally compact uniform spaces.

A simple, yet beautiful, example is the open unit disk in the plane. In terms ofits Euclidean geometry, each pair of points can be joined by a twisted double conewhich stays away from the boundary and is not much bigger than the distancebetween the given points (it is a uniform space). On the other hand, the diskalso admits a non-euclidean geometry, in terms of its Poincare hyperbolic metric,and as such the disk is a Gromov hyperbolic space.

The Bonk, Heinonen, Koskela theory asserts that this phenomenon holds in avery general setting. The complete proof of their result is presented in Chapters2-5 of [BHK01] and beyond the scope of our discussion. However, there are twobasic results involved which are central to my workshop lectures: Fact 4.1 saysthat every locally compact uniform space has a Gromov hyperbolic quasihyper-bolization; Fact 5.1 says that every (proper geodesic) Gromov hyperbolic spacecan be uniformized. I will describe what uniform spaces are, what their con-nection is with Gromov hyperbolicity, and explain some of the ideas behind theproofs. Time permitting, I will also look at the related question of when thereexists a uniformization with the property that the associated measure (see (2.7))has regular volume growth. My conference lecture will focus on §6.D and §6.E.

For the remainder of this introduction, I advertise results from [Her04] and[Her06] hoping to wet the reader’s appetite for this flavor of metric measure spacegeometric function theory. See §2-§5 for precise definitions.

In [BHR01] and [BKR98] the authors investigate conformal deformations ofthe unit ball in Euclidean space. The primary object of study in these notesis the geometry of quasiconformal deformations of an abstract metric measurespace (Ω, d, µ). Following BHKR, we consider a metric-density ρ on Ω and Ωρ =(Ω, dρ, µρ) denotes the deformed space (see subsection 2.G). We are interestedin the situation when this new space Ωρ is uniform (see Section 3) and describethis by calling such a ρ a uniformizing density. Every proper geodesic Gromovhyperbolic space can be uniformized, and, there is a natural canonical propergeodesic space associated with any locally compact abstract domain, namely, itsquasihyperbolization; see Facts 4.1 and 5.1. However, in general the associatedmeasure (see (2.7)) may fail to have Ahlfors regular volume growth. For example,applying the BHK uniformization to the quasihyperbolized Euclidean unit ballwe obtain a new metric measure space which has exponential volume growth.

The theory developed in [BHK01] is exploited in [Her06] to extend some resultsof [BHR01] to the setting of abstract metric measure spaces (Ω, d, µ). Moreimportantly, we establish the result given below which provides an answer to the


question: When does an abstract domain admit a quasiconformal deformationwhich is both uniformizing and such that the induced measure (2.7) satisfies thenatural volume growth estimate? That is, when is there a conformal uniformizingdensity? In particular, the induced measure should be Ahlfors regular. Undercertain reasonable minimal hypotheses, this occurs precisely when the conformalAssouad dimension of the space’s Gromov boundary is small enough. See §5.Efor a discussion of the proof of the following.

Theorem A. Let Ω be an abstract domain with bounded Q-geometry. Suppose Ωadmits a bounded uniformizing conformal density. Then Ω has a Gromov hyper-

bolic roughly starlike quasihyperbolization and the conformal Assouad dimension

of its Gromov boundary is strictly less than Q. The converse holds too, provided

we assume that the Gromov boundary of Ω is uniformly perfect.

The above result is quantitative: the asserted constants depend only on the dataassociated with Ω and the density.

In what follows we consider metric measure spaces (Ω, d, µ) which satisfy thefollowing basic minimal hypotheses:

Ω is an abstract domain having bounded Q-geometry and aGromov hyperbolic roughly starlike quasihyperbolization.

Precise definitions are stated in subsections 2.B, 2.D, 2.H, 5.C; roughly, thesehypotheses ensure that Ω has ‘enough’ of the local properties enjoyed by domainsin Euclidean space. The data associated with these basic hypotheses consists ofsix parameters: Q (the ‘dimension’), M , m, λ (the bounded geometry constants),δ (the Gromov hyperbolicity constant) and κ (the rough starlike constant).

There are a number of auxiliary results (namely, Theorems B-F) needed forthe proof of Theorem A; all of these can be found in [Her04] or [Her06]. Firstwe have the so-called Gehring-Hayman Inequality (cf. [GH62]); it is an essentialtool for most of what follows. This was proved in [BKR98, Theorem 3.1] fordeformations of the Euclidean unit ball and in [HR93] for quasiconformal imagesof uniform domains in Euclidean space; see also [BB03, Theorem 2.3], [BHK01,Chpt. 5] and [HN94]. Our proof of the following (see [Her04, Theorem A]) utilizesideas from both [BKR98, Theorem 3.1] and [HR93, Theorem 1.1].

Theorem B. Let ρ be an Ahlfors Harnack density on a uniform Loewner met-

ric measure space (Ω, d, µ). Then there exists a constant Λ such that for all

quasihyperbolic geodesics [x, y]k with endpoints in Ω,

ℓρ ([x, y]k) ≤ Λ dρ(x, y).

This result is quantitative: Λ depends only on the data associated with Ω.Throughout this article the symbol Λ will stand for this Gehring-Hayman In-equality constant.

Here is a simple, but useful, consequence of the Gehring-Hayman Inequality:if there is an arc α joining some point w in Ω to some point ζ in ∂Ω with


ℓρ(α) <∞, then ℓρ(γ) <∞ for every quasihyperbolic geodesic ray going to ζ. Infact, there is even a ‘radial limit theorem’ [Her04, Theorem B] which says thatthis is true for modQ-a.e. point of ∂Ω.

Next we communicate the primary tool employed in our proof of Theorem A.It is based on a lifting procedure discussed in [BKR98, 2.7] and established forthe Euclidean unit ball as [BHR01, Proposition 1.25]. See (5.9) and (2.5) for thedefinitions of ρν (the lift of ν) and δν,1/P (the quasimetric determined by ν). See§5.D for a discussion of the proof of the following.

Theorem C. Assume the basic minimal hypotheses, that the Gromov boundary

of Ω is uniformly perfect, and that P < Q. Suppose ν is a P -dimensional metric

doubling measure on ∂GΩ. Then the lift ρ = ρν of ν is a doubling conformal

density on Ω and the natural map (∂ρΩ, dρ) → (∂GΩ, δν,1/P ) is bilipschitz.

The Bonk-Heinonen-Koskela uniformization theory is a crucial tool employedin all our arguments and permits us to replace the space Ω with a boundeduniform space Ωε where the geometry is more transparent; see Fact 5.1. A keyingredient in our proof of Theorem A is the following generalization of [BHR01,Proposition 2.11]. In particular, it asserts that a conformal density on a boundeduniform space is uniformizing if and only if the associated measure (2.7) is adoubling measure on the original space. (See §5.E for the precise definition of adoubling conformal density.)

Theorem D. Assume the basic minimal hypotheses. Let Ωε be any BHK-

uniformization of Ω. Suppose ρ is a conformal density on Ω. Then the following

are quantitatively equivalent:

(a) ρ is doubling on Ω.

(b) Ωρ is bounded and uniform.

(c) Ωρ is bounded and Q-Loewner.

(d) Ωρ is bounded, Q-Loewner and Ahlfors Q-regular.

(e) the identity map Ωρ → Ωε is quasisymmetric.

Again, this result is quantitative: the asserted constants depend only on the dataassociated with Ω and ρ, and the related data. Also, we point out that the proofof (b) shows that the quasihyperbolic geodesics in Ω will be uniform arcs in Ωρ.

A crucial component of the proof of Theorem C is the following result whichpermits us to estimate dρ(x) = distρ(x, ∂ρΩ) in terms of ρ(x)d(x). More precisely,it tells us that Ahlfors Harnack metric-densities are Koebe under the right condi-tions. The lower bound is immediate via the Harnack inequality. To obtain anyupper bound, we at least need ∂ρΩ 6= ∅. In fact, we require a condition whichensures that Ω has a uniformly thick boundary as seen from each point. Withthis in mind, we introduce the following notion: we say that (Ω, d, µ) satisfies aWhitney ball modulus property if there exists a constant m > 0 such that

modQ(B(x;λ d(x)), ∂Ω; Ω) ≥ m for all x ∈ Ω.


Theorem E. Let ρ be a Ahlfors Harnack metric-density on a uniform Loewner

abstract domain (Ω, d, µ). Suppose that Ω enjoys a Whitney ball modulus prop-

erty. Then there is a constant K such that for all x ∈ Ω,

K−1ρ(x)d(x) ≤ dρ(x) ≤ Kρ(x)d(x);

the constant K depends only on the data associated with Ω.

An important consequence of Theorem E is that the quasihyperbolizationsof Ω and Ωρ are bilipschitz equivalent, and it follows that (Ωρ, kρ) is a Gromovhyperbolic space.

We mention that any uniform Loewner space with connected boundary satis-fies a Whitney ball modulus property, provided it and its boundary are simulta-neously bounded or unbounded. Similarly any bounded uniform Loewner spacewith a finite number of non-degenerate boundary components will enjoy thismodulus property. Here is a sufficient condition for this property to hold whichallows for a totally disconnected boundary.

Theorem F. Let (Ω, d, µ) be a locally Loewner, uniform metric measure space.

Assume Ω and ∂Ω are either both bounded or both unbounded. Suppose that for

some p > 0, ∂Ω satisfies the Hausdorff p-content condition

Hp∞(∂Ω ∩ B(ζ; r)) ≥ c r

p for all 0 < r ≤ diam(∂Ω) and all ζ ∈ ∂Ω.

Then Ω enjoys a Whitney ball modulus property with a constant m which depends

only on c and the data associated with Ω.

In contrast to the Euclidean case, the converse to the above is false; see[Her04, Example 3.2] which furnishes a space with an isolated boundary pointwhich nonetheless satisfies a Whitney ball modulus property.

Our notation is relatively standard and, for the most part, conforms with thatof [BHK01]. We write C = C(a, . . .) to indicate a constant C which depends onlyon the parameters a, . . .; the notation A . B means there exists a finite constantc with A ≤ cB, and A ≃ B means that both A . B and B . A hold. Typicallya, b, c, C,K, . . . will be constants that depend on various parameters, and we tryto make this as clear as possible often giving explicit values, however, at timesC will denote some constant whose value depends only on the data present butmay differ even on the same line of inequalities.

2. Metric Space Background

Naturally there are scores of references for metric space geometry. Here is abrief list of some texts which I have found especially helpful: [BH99], [BBI01],[Hei01], [Sem01], [Sem99], [DS97], and of course the references mentioned in theseworks.


2.A. General Information. In what follows (X, d) will always denote a genericmetric space possessing no additional presumed properties. For the record, thismeans that d is a distance function; that is, d : X × X → R is positive semi-definite, symmetric, and satisfies the triangle inequality. We often write thedistance between x and y as d(x, y) = |x− y|. The open ball (sphere) of radius rcentered at the point x is B(x; r) := y : |x−y| < r (S(x; r) := y : |x−y| = r).When B = B(x; r) and λ > 0, λB := B(x;λ r). We say that X is a proper met-ric space if it has the Heine-Borel property that every closed ball is compact (orequivalently, the compact sets are exactly the closed and bounded sets).

In general, we work in the setting of a metric measure space (X, d, µ) withX a non-complete locally complete (often locally compact) rectifiably connectedmetric space and µ a Borel regular measure satisfying µ[B(x; r)] > 0 for eachball.

Recall that every metric space can be isometrically embedded into a completemetric space. We let X denote the metric completion of a metric space X andwe call ∂X = X \X the metric boundary of X. Then d(x) = dist(x, ∂X) is thedistance from a point x ∈ X to the boundary ∂X of X; note that when ∂X isclosed in X, we have d(x) > 0 for all x ∈ X. For example, this holds when X

is locally compact. Of course, if X is complete to begin with, then ∂X = ∅ andd(x) = ∞ for all x ∈ X. We call X locally complete provided d(x) > 0 for allx ∈ X.

In a locally complete metric space we make extensive use of the notation

B(x) := B(x; d(x)).

In this setting, we call λB(x) = B(x;λ d(x)) a Whitney ball in X with associatedWhitney ball constant λ ∈ (0, 1).

It is convenient, at times, to consider quasimetric spaces (X, q). We call q aquasimetric on X if q : X ×X → R is symmetric and positive definite but onlysatisfies

q(x, y) ≤ K (q(x, z) + q(y, z)) for all x, y, z ∈ X

in place of the triangle inequality. See [Hei01, 14.1], [Sem01] and [DS97].

Starting with a quasimetric q, there is a standard way to define a pseudometricd with d ≤ q (cf. [BH99, 1.24, p.14]), but it may happen that d(x, y) = 0 forsome x 6= y. However, by first ‘snowflaking’ q and then applying this procedurewe can arrive at an honest distance function; see [Hei01, Proposition 14.5] or[BH99, Proposition 3.21, p.435].

2.1. Fact. Let q be a quasimetric on X. There is an ε0 > 0 depending only on

the quasimetric constant K for q such that for all ε ∈ (0, ε0), the quasimetric

qε(x, y) = q(x, y)ε is bilipschitz equivalent to an honest distance function d on

X; in fact there is a constant L = L(ε,K) such that

L−1qε(x, y) ≤ d(x, y) ≤ qε(x, y) for all x, y ∈ X.


We remark that all the quasimetrics qε as defined above are QS equivalent toeach other.

Another useful notion, apparently introduced by Vaisala, is that of a meta-

metric m : X ×X → R which is symmetric, non-negative, satisfies the triangleinequality, but only

m(x, y) = 0 =⇒ x = y

and so possibly m(x, x) > 0. See [Vai05a, 4.2] for a treatment of metametricspaces.

A metric space X is called uniformly perfect provided it has at least twopoints and there is a constant ϑ ∈ (0, 1) such that for all balls B ⊂ X, B \ϑB 6= ∅ provided X \ B 6= ∅. This concept, which involves three points, isespecially useful when dealing with quasisymmetric maps and also with doublingmeasures (see §2.F). The property of being uniformly perfect is preserved byquasisymmetric homeomorphisms, with the new constant depending only on theoriginal constant and the quasisymmetry data; in particular, one can ask whetheror not a conformal gauge is uniformly perfect (see §2.C).

It is a routine exercise to see that uniformly perfect locally compact spacescontain quasisymmetrically embedded middle-third Cantor dusts. Using thisfact, together with a scaling argument and properties of quasisymmetric homeo-morphisms (e.g. [Hei01, 11.10,11.11]), one can verify a version of the following.For a simple more direct approach, which also provides the indicated explicitconstants, see [Her06, Lemma 4.2].

2.2. Fact. Suppose X is a uniformly perfect compact metric space. Then X

satisfies the p-dimensional Hausdorff measure density condition

Hp[B(x; r)] ≥ rp

6for all 0 < r ≤ diam(X) and all x ∈ X,

where p = 1/ log2(4/ϑ) and ϑ is the uniform perfectedness constant.

The above result can be used in conjunction with Theorem F to see that theWhitney ball modulus property holds.

2.B. Abstract Domains. We call a metric measure space (Ω, d, µ) an abstract

domain if Ω is a non-complete locally complete rectifiably connected metric space(and µ a Borel regular measure with dense support). An important example ofsuch a space is, of course, a proper subdomain of Euclidean space with eitherEuclidean distance or the induced Euclidean length distance.

Unless explicitly indicated otherwise, the adjective locally means that themodified property or condition holds in all Whitney-type balls λB(x) where0 < λ < 1 is some fixed constant which we call the Whitney ball constant;when there are several such local conditions in play, we always take λ to be theminimum of all the associated Whitney ball constants.


2.C. Maps and Gauges. An embedding f : X → Y from a metric space Xinto a metric space Y is quasisymmetric, abbreviated QS, if there is a homeomor-phism η : [0,∞) → [0,∞) (called a distortion function) such that for all triplesx, y, z ∈ X,

|x− y| ≤ t|x− z| =⇒ |fx− fy| ≤ η(t)|fx− fz|.These mappings were studied by Tukia and Vaisala in [TV80]; see also [Hei01].The bilipschitz maps form an important subclass of the quasisymmetric maps;f : X → Y is bilipschitz if there is a constant L such that for all x, y ∈ X,

L−1|x− y| ≤ |fx− fy| ≤ L|x− y|.

More generally, a map f : X → Y is an (L,C)-quasiisometry if L ≥ 1, C ≥ 0and for all x, y ∈ X,

L−1|x− y| − C ≤ |fx− fy| ≤ L|x− y| + C.

There seems to be no universal agreement regarding this terminology; some au-thors use the adjective quasiisometry to mean what we have called bilipschitz,and then a rough quasiisometry satisfies our definition of quasiisometry. So thereader should beware! Of course a (1, 0)-quasiisometry is simply called an isom-

etry (onto its range).

Note that the above definitions also make sense for mappings of quasimetricspaces.

Given a metric (or a quasimetric) on X, we can form the conformal gauge

G on X consisting of all metrics on X which are QS equivalent to the original(quasi)metric. That is, G is the family of all metrics ∂ on X such that the identitymap (X, d) → (X, ∂) is QS. See [Hei01, Chapter 15] for more discussion of thistopic.

An embedding f : X → Y from a metric space X into a metric space Y iscalled quasimobius, abbreviated QM, if there is a homeomorphism ϑ : [0,∞) →[0,∞) (called a distortion function) such that for all quadruples x, y, z, w ofdistinct points in X,

|x, y, z, w| ≤ t =⇒ |fx, fy, fz, fw| ≤ ϑ(t)

where the absolute cross ratio is

|x, y, z, w| =|x− y||z − w||x− z||y − w| .

These mappings were introduced and investigated by Vaisala in [Vai85]; seealso [Vai05a]. Every QS homeomorphism is QM; the converse holds in certainspecial cases. Clearly Mobius transformations are QM maps in Euclidean space;however, a Mobius transformation from the unit ball onto a half-space is not QS.The QM maps are more flexible than the QS.

The QS and QM maps are defined by global conditions whereas QC (quasicon-formal) maps only satisfy a local condition. I highly recommend Tyson’s recentsurvey article [Tys03]. Vaisala’s notes [Vai71] are the classical reference for QCmaps in the Euclidean setting. These maps have been studied in the Heisenberg


group setting and there is still much research underway there. Heinonen andKoskela strongly advanced the theory in the general metric space setting; see[HK95] and [HK98]. See Koskela’s notes [Kos07] for a ‘modern’ approach to QCmaps in the Euclidean setting.

There are three so-called definitions for QC maps: the metric definition, thegeometric definition, and the analytic definition. We present the first two. Ahomeomorphism f : X → Y is (metrically) quasiconformal provided there is aconstant H <∞ such that for all x ∈ X,

lim suprց0

H(x, f, r) ≤ H where H(x, f, r) =L(x, f, r)

l(x, f, r),

L(x, f, r) = sup|f(y) − f(x)| : |x− y| ≤ r ,l(x, f, r) = inf|f(y) − f(x)| : |x− y| ≥ r .

A homeomorphism f : X → Y is (geometrically) quasiconformal provided thereis a constant K <∞ such that for all curve families Γ in X,

K−1 mod(Γ) ≤ mod(fΓ) ≤ K mod(Γ).

Notice that unlike the metric definition, which makes sense for any pair of met-ric spaces, the geometric definition requires measure metric spaces. These aregenerally assumed to be Ahlfors Q-regular spaces (see §2.J) in which case mod(·)denotes the Q-modulus.

2.D. Length and Geodesics. The length of a continuous path γ : [0, 1] → X

is defined in the usual way by

ℓ(γ) := supn∑

i=1

|γ(ti) − γ(ti−1)| where 0 = t0 < t1 < · · · < tn = 1.

We call γ rectifiable when ℓ(γ) < ∞. We let Γ(x, y) = Γ(x, y;X) denote thecollection of all rectifiable paths joining x and y in X; in general we shouldalso indicate the metric in this notation, but it will always be understood fromcontext. Vaisala’s notes [Vai71, §1-§5] provide an excellent reference for studyingproperties of curves, and the results are valid in the general metric space setting.Each rectifiable path γ : [0, 1] → X has an associated arclength function s :[0, 1] → [0, ℓ(γ)], given by s(t) = ℓ(γ[0, t]), which is of bounded variation. Givena Borel measurable function ρ : X → [0,∞], we define

∫

γ

ρ ds :=

∫1

0

ρ(γ(t)) ds(t).

An arc in a metric space X is the homeomorphic image of an interval I ⊂ R.Given two points x and y on an arc α, we write α[x, y] to denote the subarc ofα joining x and y.

A geodesic in X is the image ϕ(I) of some isometric embedding ϕ : I → X

where I ⊂ R is an interval; we use the adjectives segment, ray, or line (re-spectively) to indicate that I is bounded, semi-infinite, or all of R. When ϕ is


L-bilipschitz we call ϕ(I) an L-quasigeodesic. More generally, if ϕ is (L,C)-quasiisometric, then we call ϕ(I) an (L,C)-quasigeodesic. Thus γ is an L-quasigeodesic precisely when

∀x, y ∈ γ : ℓ(γ[x, y]) ≤ L|x− y|;classically, such curves in the plane R

2 were called chord arc curves.

A metric space is geodesic if each pair of points can be joined by a geodesicsegment. We use the notation [x, y] to mean a (not necessarily unique) geodesicsegment joining points x, y; such geodesics always exist if our space is geodesic,but may not be unique. (If there is some other distance function, such as k, thenwe write [x, y]k to denote a k-geodesic joining x, y). We consider a given geodesic[x, y] as being ordered from x to y (so we can use phrases such as the ‘first’ pointencountered). An unbounded metric space is roughly κ-starlike with respect to abase point w if each point lies within distance κ of some geodesic ray emanatingfrom w.

The geodesic boundary ∂gX of an unbounded geodesic metric space X is theset of equivalence classes of geodesic rays in X where two such rays are consid-ered equivalent when they are at a finite Hausdorff distance from each other.Equivalently, if α, β : [0,∞) → X are geodesic rays in X, then α ≃ β ifsupt |α(t) − β(t)| < ∞. The geodesic boundary of R

n is the sphere Sn−1. The

geodesic boundary of hyperbolic n-space (Bn, h) is also the sphere S

n−1.

Every metric space (X, d) admits a natural (or intrinsic) metric, the so-calledlength distance given by

l(x, y) := infℓ(γ) : γ a rectifiable curve joining x, y in Ω.A metric space (X, d) is a length space provided d(x, y) = l(x, y) for all pointsx, y ∈ X; it is also common to call such a d an intrinsic distance function. Noticethat an l-geodesic [x, y]l is a shortest curve joining x and y.

The Hopf-Rinow Theorem (see [Gro99, p.9], [BBI01, p.51], [BH99, p.35]) saysthat every locally compact length space is proper (and therefore geodesic). Ina general length space, when geodesics may not exist, one works with so-calledshort arcs; see [Vai05a].

Since |x−y| ≤ ℓ(x, y) for all x, y, the identity map (X, l)id→ (X, d) is Lipschitz

continuous. It is important to know when this map will be a homeomorphism(cf. [BHK01, Lemma A.4, p.92]). Notice that the identity map (X, d) → (X, l) isuniformly locally Lipschitz when X is locally quasiconvex; see §2.E. More gen-erally, one can show that the identity map (X, l) → (X, d) is a homeomorphismprecisely when X satisfies a weak notion of local quasiconvexity; see [BH07].

2.E. Connectivity Conditions. A metric space (X, d) is a-quasiconvex pro-vided each pair of points can be joined by a path whose length is at most atimes the distance between its endpoints. A locally complete space X is locally

quasiconvex if there exists a constant a ≥ 1 such that for all z ∈ X, pointsx, y ∈ λB(z) can be joined by a rectifiable arc α in X with ℓ(α) ≤ a|x− y|; we


abbreviate this by the phrase ‘X is locally a-quasiconvex’. (Here it is understoodthat there is some Whitney ball constant λ ∈ (0, 1) which may also depend onother parameters).

A space (X, d) is c-linearly locally connected, or c-LLC, if c ≥ 1 and thefollowing conditions hold for all x ∈ X and all r > 0:

points in B(x; r) can be joined in B(x; c r)(LLC1)

and

points in X \ B(x; r) can be joined in X \ B(x; r/c).(LLC2)

Here the phrase ‘can be joined’ means ‘can be joined by a continuum’. We alsouse the term LLC with respect to arcs in which case ‘can be joined’ means ‘canbe joined by a rectifiable arc’. Note that quasiconvexity implies LLC1 (even withrespect to arcs).

The generic example of a space which does not satisfy the LLC2 condition isthe interior of an infinite Euclidean cylinder such as B

n−1 × R ⊂ Rn. However,

for 2 ≤ k < n the regions Bn−k × R

k ⊂ Rn are easily seen to be 1-LLC2. The

complement of a semi-infinite slab (e.g., Rn \ (x1, . . . , xn) : x1 ≥ 0, |xn| ≤ 1)

fails to be LLC1.

Ahlfors regular Loewner spaces are LLC; see [HK98, Theorem 3.13]. Uniformdomains also enjoy this property, but not necessarily uniform spaces. The LLCcondition was invented by Gehring who first used it to characterize quasidisks;see [Geh82] and the references mentioned therein.

2.F. Doubling and Dimensions. The p-dimensional Hausdorff measure of aset A ⊂ X is given by Hp(A) := limr→0 Hp

r(A) where

Hpr(A) := inf

∑diam(Bi)

p : A ⊂ ∪Bi, Bi balls with diam(Bi) ≤ r.

The Hausdorff p-content of A is just Hp∞(A). The Hausdorff dimension of A is

determined by

dimH(A) := inf p > 0 : Hp(A) = 0 .We also require the Assouad dimension of X which is given by

dimA(X) := infp : #S ≤ C(R/r)p for all S ⊂ X

with r ≤ |x− y| ≤ R for all x, y ∈ Swhere #S denotes the cardinality of the set S. See [Hei01, 10.15] or [Luu98,3.2]. The spaces with finite Assouad dimension are precisely the doubling spaces(which we discuss below in more detail). Finally, the conformal Assouad dimen-

sion of a metric space X is

c-dimA(X) := infdimA(X, d) : d ∈ G,where G is the conformal gauge on X determined by the original metric; see[Hei01, 15.8, p.125].


A metric space (X, d) satisfies a (metric) doubling condition if there is a con-stant N such that each ball in X of radius R can be covered by at most N balls ofradius R/2; these are precisely the spaces of finite Assouad dimension. A Borelmeasure ν is a doubling measure on X if there is a constant D = Dν such that

ν[B(x; 2r)] ≤ Dν[B(x; r)] for all x ∈ X and all r > 0.

A Borel measure ν on X is p-homogeneous if there is a constant C = Cν suchthat

ν[B(x;R)]

ν[B(x; r)]≤ C

(R

r

)p

for all x ∈ X and all 0 < r ≤ R.

Obviously every homogeneous measure is doubling; the converse holds too withC = D and p = log

2(D). Every Ahlfors Q-regular measure is Q-homogeneous.

The existence of a doubling measure is easily seen to imply a metric doublingcondition; the converse holds if our metric space is complete. Here is a precisestatement of this result, which is due to Vol’berg and Konyagin for compactspaces, and Luukkainen and Saksman for complete spaces (see [Hei01, Theo-rem 13.5]).

2.3. Fact. A complete doubling space X carries a p-homogeneous measure for

each p > dimA(X).

An especially important property of doubling measures is their exponentialdecay on uniformly perfect spaces, which we record as follows; see [Hei01, (13.2)]or [Sem99, Lemma B.4.7, p.420].

2.4. Fact. Let ν be a doubling measure on a uniformly perfect metric space.

There are constants C ≥ 1 and α > 0, depending only on the doubling constant

for ν and the uniformly perfect constant, such that for all balls B(z; r) ⊂ B(x;R),

ν[B(z; r)]

ν[B(x;R)]≤ C

(r

R

)α

.

Now we discuss an interesting way to deform the geometry of a doubling space.Let ν be a doubling measure on a metric space (X, d). For each α > 0 we defineδ = δν,α by

(2.5) δ(x, y) := ν[B(xy)]α, where B(xy) := B(x; |x− y|) ∪ B(y; |x− y|);see [DS97, §16.2], [Sem99, (B.3.6)], [Hei01, 14.11]. This always defines a quasi-metric on X, and, when X is uniformly perfect, the identity map (X, d) → (X, δ)will be quasisymmetric and (X, δ, ν) will be Ahlfors (1/α)-regular. Moreover,there is an α0 > 0 (depending only on the doubling constant for ν) such thatfor all 0 < α < α0, δν,α is bilipschitz equivalent to an honest distance functionon X (see Fact 2.1). In particular, if ν is p-homogeneous, then δν,1/p is alreadybilipschitz equivalent to an honest distance function (e.g., if (X, d, ν) is Ahlforsp-regular, then δν,1/p is bilipschitz equivalent to d).

In conjunction with the above chain of ideas, we declare ν to be a p-dimensional

metric doubling measure on X if ν is a doubling measure on X with the prop-erty that δν,1/p is bilipschitz equivalent to a distance on X. For example, a


p-homogeneous measure will be a p-dimensional metric doubling measure. Wesummarize the above comments; see [Hei01, 14.11,14.14], [Sem99, B.3.7, B.4.6,p.421], [DS97, 16.5,16.7,16.8].

2.6. Fact. Let ν be a p-dimensional metric doubling measure on a uniformly

perfect metric space (X, d). Define δ = δν,1/p as in (2.5). Then δ is a quasimetric

on X which is bilipschitz equivalent to a distance function on X, the identity map

(X, d) → (X, δ) is quasisymmetric, and (X, δ, ν) is an Ahlfors p-regular space.

All of the new parameters depend only on the original data for X and ν.

There is one final comment we wish to point out regarding the quasimetricsδν,α. As above, suppose ν is a doubling measure on a metric space (X, d), andsuppose X has another metric, say, ∂ which is QS equivalent to d. Then byusing the doubling property of ν in conjunction with quasisymmetry we see thatν[Bd(xy)] ≃ ν[B∂(xy)] (where these sets are defined as above using balls centeredat x and y in the appropriate metrics); here the constant depends only on thedoubling constant and the quasisymmetry data. It therefore follows that thequasimetric δd (defined as in (2.5) via Bd(xy)) is bilipschitz equivalent to δ∂

(defined via B∂(xy)).

We note the important fact that quasisymmetric homeomorphisms preservethese doubling conditions; cf. [Hei01, Theorem 10.18] or [DS97, Lemma 16.4].In particular, the notions of doubling measure, the quasimetrics δν,α, and metricdoubling measures do not depend on the given distance function per se; they allmake sense for a conformal gauge.

2.G. Quasiconformal Deformations. Given an abstract domain (Ω, d, µ) anda positive Borel measurable function ρ on Ω, we wish to define a new metric mea-sure space Ωρ = (Ω, dρ, µρ) which is a quasiconformal deformation of Ω. (Abovein §2.F we described another method for deforming the geometry of Ω which wasbased on having a doubling measure. See Fact 2.6.)

We start by defining the ρ-length of a rectifiable curve γ via

ℓρ(γ) :=

∫

γ

ρ ds

and then the ρ-distance between two points x, y is

dρ(x, y) := infℓρ(γ) : γ a rectifiable curve joining x, y in Ω;see §2.D. The careful reader no doubt recognizes that, in general, dρ(x, y) couldbe zero or even infinite; in order to ensure that dρ be an honest distance function,we must require that 0 < dρ(x, y) < ∞ for all points x, y ∈ Ω. We designatethis by calling such ρ a metric-density on Ω. One way to guarantee this is to askthat ρ be locally bounded away from zero and infinity. In practice, our densitieswill always satisfy a Harnack inequality—see below—so this is never a problemfor us.


The ρ-balls (etc.) are written as Bρ(x; r); these are the metric balls in Ωρ, soBρ(x; r) = y ∈ Ω : dρ(x, y) < r. We define a new measure µρ by

(2.7) µρ(E) :=

∫

E

ρQdµ.

Here Q is usually the Hausdorff dimension of (Ω, d).

When Ωρ is non-complete (which will often be the case for us), we can form∂ρΩ = Ωρ \ Ωρ and define dρ(x) = distρ(x, ∂ρΩ). In this setting we also employthe notation Bρ(x) = Bρ(x; dρ(x)); thus λBρ(x) is a Whitney ball in Ωρ.

We are especially interested in the metric-densities ρ for which Ωρ is a uniformspace, and we call such a ρ a uniformizing density (which implicitly includesthe hypothesis that Ωρ is non-complete). The Bonk-Heinonen-Koskela theoryproduces uniformizing densities on proper geodesic Gromov hyperbolic spaces;see Fact 4.1. Some other classes of metric-densities which we wish to single outfor attention include Harnack, Ahlfors, and Koebe densities; their definitionsfollow below. We let Hρ, Aρ, Kρ denote the parameters associated with thesedensities.

Before delving into the technical definitions, we wish to make a few com-ments. The reader no doubt has encountered deformations of Euclidean domainsΩ ⊂ R

n by continuous densities ρ; in this setting Ωρ is a conformal deformationof Ω, meaning that the identify map Ω → Ωρ is conformal (i.e., metrically 1-quasiconformal). However, in our more general setting, even for the case ρ = 1say, the identity map Ω → Ωρ = Ωl may fail to be quasiconformal (e.g., if Ω doesnot satisfy some sort of local quasiconvexity condition). A similar phenomenonholds for Borel metric-densities, even for domains Ω ⊂ R

n. Nonetheless, whenΩ is locally quasiconvex and ρ is a Harnack metric-density, Lemma 2.8 belowreveals that the identity map Ω → Ωρ is QC (and according to Proposition 2.9even QS under the right circumstances). This is a good thing: we want Ωρ to bea quasiconformal deformation of Ω.

With this in mind, we pronounce the following definitions. First, we declare ρto be a bounded density if the deformed space Ωρ is bounded, i.e., diamρ(Ω) <∞.

Next, we call ρ a Harnack density provided it satisfies a uniform local Harnacktype inequality: for all points x in Ω,

(H)1

H≤ ρ(y)

ρ(x)≤ H for all y ∈ λB(x).

HereH = Hρ ≥ 1 and 0 < λ < 1 (generally λ will be small). Note that in contrastto the situation in [BKR98, p.637], the validity of (H) for some 0 < λ < 1 neednot mean a similar set of inequalities will hold for λ = 1/2. The condition (H)provides local control and permits the use of standard chaining type arguments;e.g., see Lemma 2.13.

We call ρ an Ahlfors density if the associated metric measure space Ωρ =(Ω, dρ, µρ) is Ahlfors upper Q-regular (cf. §2.J); i.e., if µρ satisfies a global upper


Ahlfors Q-regular volume growth estimate: there is a constant A = Aρ such thatfor all points x in Ω,

(A) µρ[Bρ(x; r)] ≤ ArQ for all r > 0.

The positive real number Q is generally the Hausdorff dimension of our space;it must agree with the number Q appearing in the definition of a Loewner space(a notion also discussed in §2.J). The volume growth condition (A) ensuresthat Ωρ satisfies an upper mass condition and so provides modulus estimates viaFacts 2.15, 2.16, 2.17.

Below (in §2.H) we discuss the density 1/d which determines the quasihyper-bolic distance; of course this is a continuous Harnack density, but in general 1/ddoes not satisfy the volume growth requirement (A).

We call ρ a Koebe density if Ωρ is non-complete and there is a constantK = Kρ

such that dρ(x) = distρ(x, ∂ρΩ) enjoys the property

(K)1

K≤ dρ(x)

ρ(x)d(x)≤ K for all x ∈ Ω.

(Note that when ρ is a Harnack density, dρ(x) ≥ (λ/H)ρ(x)d(x) always holds,and so it is the upper estimate which is needed.) For example, if ρ = |f ′| where fis a holomorphic homeomorphism defined in a subdomain Ω of the complex plane,then a classical theorem in univalent function theory asserts that ρ is a Koebedensity with constant K = 4. As another example we note that Theorem Easserts that any Harnack Ahlfors density on a uniform Loewner space (with suf-ficiently ‘thick’ boundary) is a Koebe density; see [Her04, Theorem E]. We pointout that when ρ is a Koebe density on (Ω, d), the identity map (Ω, k) → (Ωρ, kρ)is easily seen to be Kρ-bilipschitz; here (Ωρ, kρ) denotes the quasihyperbolizationof Ωρ.

We employ the terminology conformal density for a metric-density which isHarnack, Ahlfors, and Koebe. A basic example of a conformal density is ρ = |f ′|for any holomorphic homeomorphism |f ′| defined in a subdomain of the complexplane; we refer to [BKR98, Section 2] for other examples of conformal densitieson the Euclidean unit ball. The reader should be aware that the phrase ‘ρ is aconformal density’ does not necessarily mean that the identity map Ω → Ωρ isquasiconformal (unless Ω is locally quasiconvex).

Here are some especially useful estimates which also provide information con-cerning the identity map Ω → Ωρ for certain densities. Roughly speaking, thismap is locally bilipschitz (therefore quasiconformal) for Harnack densities anduniformly locally quasisymmetric for Harnack Koebe densities, provided Ω is lo-cally quasiconvex. Proposition 2.9 gives a significant strengthening of this result.

2.8. Lemma. Let ρ : Ω → (0,∞) be a Harnack density on a locally a-

quasiconvex abstract domain (Ω, d). Put η = λ/2a. Then for all z ∈ Ω,

1

H≤ dρ(x, y)

ρ(z)|x− y| ≤ aH for all points x 6= y in ηB(z);


in particular,

1

H≤ diamρ[ηB(z)]

ηρ(z)d(z)≤ 2aH.

If ρ is also a Koebe density, then for all 0 ≤ ϑ ≤ λ/2C and all x ∈ Ω,

C−1ϑB(x) ⊂ ϑBρ(x) ⊂ CϑB(x),

where C = aHK. Here H = Hρ, K = Kρ and λ is the Whitney ball constant.

One immediate consequence of Lemma 2.8 is that the identity map Ω → Ωρ

is metrically quasiconformal with linear dilatation aH2. In addition, because of

the definition of the associated measure (see (2.7)), a straightforward calcula-tion reveals that this identity map is geometrically quasiconformal with innerdilatation H

Q and outer dilatation (aH)Q. (Here we assume a Harnack densityon a locally quasiconvex Ω.) It is therefore natural to inquire about possiblequasisymmetry properties of this identity map.

Heinonen and Koskela proved that a quasiconformal map of bounded Ahlforsregular spaces, with domain a Loewner space and a linearly locally connectedtarget space, is in fact quasisymmetric [HK98, Theorem 4.9]. The corollary tothe following analog of their result is used in the proof of Theorem D; note thathere our domain space is not assumed to be Ahlfors regular.

2.9. Proposition. Let Ω be a bounded locally quasiconvex Q-Loewner space.

Suppose ρ is a conformal density on Ω with Ωρ a bounded linearly locally con-

nected space. Then the identity map Ω → Ωρ satisfies the weak-quasisymmetry

condition

∀x, y, z ∈ Ω : |x− y| ≤ |x− z| =⇒ dρ(x, y) ≤ Ldρ(x, z)

for some constant L which depends only on the data associated with Ω, ρ, Ωρ,

and the ratios r, q given in the proof.

2.10. Corollary. Let Ω be a bounded quasiconvex Q-Loewner space. Suppose

ρ is a conformal density on Ω and Ωρ is a bounded Q-Loewner space. Then the

identity map Ω → Ωρ is quasisymmetric with a distortion function which depends

only on the data associated with Ω, ρ, Ωρ, and the ratios r, q given in the proof

of Proposition 2.9.

As an exercise to help understand the various properties of these metric-densities, the interested reader can provide a proof for the following [Her06,Lemma 2.6].

2.11. Lemma. Suppose (Ω, d, µ) is a locally a-quasiconvex abstract domain. Let

τ be a positive Borel function on Ω which is locally bounded away from 0 and

∞. Put ∆ = Ωτ . If σ is a metric-density on ∆, then its pull-back ρ = σ τ is a

metric-density on Ω, Ωρ = ∆σ, and

(a) ρ and σ either are, or are not, both Ahlfors regular (with Aρ = Aσ),

(b1) if σ, τ are both Koebe, then so is ρ with Kρ = KσKτ ,


(c1) if σ, τ are both Harnack, then so is ρ with Hρ = HσHτ .

On the other hand, if ρ is a metric-density on Ω, then its push-forward σ = ρ τ−1

is a metric-density on ∆, ∆σ = Ωρ, (a) holds, and

(b2) if ρ, τ are Koebe, then so is σ with Kσ = KρKτ ,

(c2) if ρ, τ are Harnack and τ is Koebe, then σ is Harnack with Hσ = HρHτ .

2.H. Quasihyperbolic Distance and Geodesics. The quasihyperbolic dis-tance in an abstract domain (Ω, d) is defined by

k(x, y) = kΩ(x, y) := inf ℓk(γ) = inf

∫

γ

ds

d(z)

where the infimum is taken over all rectifiable curves γ which join x, y in Ω.The quasihyperbolization of an abstract domain (Ω, d) is the metric space (Ω, k)obtained by using quasihyperbolic distance. It is not hard to see that (Ω, k)is complete, provided the identity map (Ω, ℓ) → (Ω, d) is a homeomorphism;see [BHK01, Proposition 2.8]. Thus by the Hopf-Rinow theorem ([Gro99, p.9],[BBI01, p.51], [BH99, p.35]), every locally compact abstract domain has a proper(hence geodesic) quasihyperbolization.

We call the geodesics in (Ω, k) quasihyperbolic geodesics ; see §2.D. Note thatwhen ρ is a Koebe density on Ω, the identity map (Ω, k) → (Ωρ, kρ) is bilips-chitz and we find that quasihyperbolic geodesics in Ω are quasihyperbolic quasi-geodesics in Ωρ; that is, a geodesic in (Ω, k) will be a quasigeodesic in (Ωρ, kρ)(the quasihyperbolization of Ωρ).

We remind the reader of the following basic estimates for quasihyperbolicdistance, first established by Gehring and Palka [GP76, Lemma 2.1]:

k(x, y) ≥ log

(1 +

ℓ(x, y)

d(x) ∧ d(y)

)≥ j(x, y) = log

(1 +

|x− y|d(x) ∧ d(y)

)≥

∣∣∣∣logd(x)

d(y)

∣∣∣∣ .

See also [BHK01, (2.3),(2.4)]. The first inequality above is a special case of themore general (and easily proved) inequality,

ℓk(γ) ≥ log

(1 +

ℓ(γ)

d(x) ∧ d(y)

)

which holds for any rectifiable curve γ with endpoints x, y.

An immediate consequence of the above inequalities is that the identity map(Ω, k) → (Ω, d) is continuous; indeed,

Bk(x;R) ⊂ (eR − 1)B(x) for all x ∈ Ω and all R > 0,

where Bk(x;R) denotes the R-ball centered at x in (Ω, k). It is important toknow when this map will be a homeomorphism (which, according to [BHK01,Lemma A.4, p.92], will be the case if and only if the identity map (Ω, ℓ) → (Ω, d)is a homeomorphism). The following provides quantitative information concern-ing this question; it is easy to verify via simple estimates for the quasihyperboliclengths of the ‘promised short arcs’.


2.12. Lemma. Suppose that (Ω, d) is a locally a-quasiconvex abstract domain.

Then for all x ∈ Ω and all R > 0,

τB(x) ⊂ Bk(x;R) provided 0 ≤ τ ≤ minλ,R/[a(1 +R)].As an exercise, the reader can check that for a domain in R

n, |x − y| ≤[d(x) + d(y)]/2 =⇒ k(x, y) ≤ 2. Thus (2/3)B(x) ⊂ Bk(x; 2).

The Harnack inequality (H), as stated in §2.G, only requires that ρ be es-sentially constant on Whitney type balls. We can do the usual chaining typearguments to see that such a density ρ will satisfy a Harnack type inequality onmuch bigger sets, of course with a change in the Harnack constant. Here is auseful example of this phenomena.

2.13. Lemma. Let ρ be a Harnack density on an abstract domain (Ω, d). If

x, y ∈ Ω satisfy k(x, y) ≤ K, then 1/H1 ≤ ρ(y)/ρ(x) ≤ H1, where H1 =H1(K,Hρ, λ).

We conclude this subsection with a covering lemma for quasihyperbolic geodesics.

2.14. Lemma. Suppose that (Ω, d) is a locally a-quasiconvex abstract domain.

Let γ be a quasihyperbolic geodesic segment or ray in Ω with endpoint x0. Let

x0, x1, x2, . . . be successive points along γ with k(xi, xi−1) = K ≤ log(1 + τ)where τ = minλ, 1/2a. Then the balls Bi = τB(xi) cover γ and have bounded

overlap:∑χBi

≤ Cχ∪Bi, where C = 1 + 4/K.

2.I. Modulus and Capacity. For p ≥ 1 we define the p-modulus of a familyΓ of curves in a metric measure space (X, d, µ) by

modp Γ := inf

∫ρ

pdµ,

where the infimum is taken over all Borel functions ρ : X → [0,∞] satisfying∫γρ ds ≥ 1 for all locally rectifiable curves γ ∈ Γ. Then the p-modulus of a pair

of disjoint compact sets E,F ⊂ X is

modp(E,F ;X) := modp Γ(E,F ;X)

where Γ(E,F ;X) is the family of all curves joining the sets E,F in X. We alsolet Γr(E,F ;X) be the subfamily of Γ(E,F ;X) consisting of the rectifiable pathsjoining E,F .

An important property is that under fairly general circumstances, modp(E,F ;X)agrees with the p-capacity of the pair E, F . There is extensive literature regard-ing these “capacity equals modulus” results; for a start, see [HK98, Proposi-tion 2.17].

For the reader’s convenience, we cite the following modulus estimates. Firstwe have the standard Long Curves Estimate; see [HK98, 3.15].

2.15. Fact. Let x ∈ X and suppose that the upper mass condition µ[B(x;R)] ≤MR

p holds for some R > 0. Let Γ be a family of curves in B(x;R) and suppose

that each γ ∈ Γ has arclength ℓ(γ) ≥ L > 0. Then

modp Γ ≤ L−pµ[B(x;R)] ≤M(R/L)p

.


Next we record the Spherical Ring Estimate; see [HK98, 3.14, p.17].

2.16. Fact. Let x ∈ X, 0 < 2r ≤ R, and suppose that the upper mass condition

µ[B(x; t)] ≤Mtp holds for all 0 < t < r +R. Then

modp(B(x; r), X \B(x;R);X) ≤ C (log(R/r))1−p,

where C = 2p+1M/ log 2.

Finally, we require the following Basic Modulus Estimate; see [BKR98, Lemma3.2].

2.17. Fact. Let (X, d, µ) be a metric measure space. Assume that ρ is a metric-

density on X whose associated measure (2.7) satisfies the Ahlfors volume growth

condition (A) at some point x ∈ E ⊂ X. Suppose that L > λ ≥ diamρE,

and that Γ is some family of curves γ in X each having one endpoint in E and

satisfying ℓρ(γ) ≥ L. Then

modQ Γ ≤ C (log (1 + L/λ))1−Q,

where C = 2Q+1A/ log 2.

2.18. Corollary. Let (X, d, µ) be a metric measure space. Assume that for

some point x ∈ E ⊂ X, the upper mass condition µ[B(x; r)] ≤ MrQ holds for

all r > 0. Suppose that Γ is a family of curves γ in X each having one endpoint

in E and satisfying ℓ(γ) ≥ L > diamE. Then

modQ Γ ≤ C (log (1 + L/ diamE))1−Q,

where C = 2Q+1M/ log 2.

In Euclidean space Rn, the n-modulus is also called the conformal modulus and

simply denoted by mod(·). Below we state some well-known geometric estimatesfor the conformal modulus mod(E,F ; Ω). Here and elsewhere in these notes,

∆(E,F ) := dist(E,F )/mindiam(E), diam(F )is the relative distance between the pair E, F of nondegenerate disjoint continua.

2.19. Facts. Let E,F be disjoint compact sets in Rn.

(a) If E,F are separated by the spherical ring B(x; s) \ B(x; t), then

mod(E,F ; Rn) ≤ ωn−1 (log(s/t))1−n.

(b) If E ∩ S(x; r) 6= ∅ 6= F ∩ S(x; r) for all t < r < s, then

mod(E,F ) ≥ σn log(s/t).

(c) If both E and F are connected, then

σn log(1 + 1/∆(E,F )) ≤ mod(E,F ; Rn) ≤ Ωn(1 + 1/∆(E,F ))n.

(d) (Comparison Principle) If A,B,E, F ⊂ Ω with A,B also compacta, then

mod(E,F ; Ω) ≥ 3−n minmod(E,A; Ω),mod(F,B; Ω), I,where I = infmod(α, β; Ω) | α ∈ Γr(E,A; Ω), β ∈ Γr(F,B; Ω).


(e) (Teichmuller Estimate) If E,F are both connected, then for all x, y ∈ E

and z, w ∈ F

mod(E,F ; Rn) ≥ τ

( |x− z||y − w||x− y||z − w|

)

where τ(r) is the capacity of the Teichmuller ring

Rn \ −1 ≤ x1 ≤ 0 or x1 ≥ r;

i.e, τ(r) = mod([−e1, 0], [re1,∞]; Rn).(f) There exists λ = λ(n) ∈ [6, 5e(n−1)/2) such that when E, F are both con-

nected and ∆(E,F ) ≥ 1,

21−nωn−1[log(λ∆(E,F ))]1−n ≤ mod(E,F ; Rn) ≤ ωn−1[log(∆(E,F ))]1−n

.

(g) (Carleman Inequality) For E ⊂ Ω,

mod(E, ∂Ω; Ω) ≥ nn−1

ωn−1 (log(|Ω|/|E|))1−n.

Here σn and ωn−1, Ωn are the spherical cap constant and the measures of the

(n− 1)-sphere, n-ball respectively.

Most of these estimates can be found in [Vai71] or [Vuo88]. Lemma 2.5 in[BH06] gives a precise formula for mod(E,F ; Rn) in the case when E,F aredisjoint closed balls.

2.J. Ahlfors Regular and Loewner Spaces. A metric measure space (X, d, µ)is Ahlfors Q-regular provided there exists a finite constant M = Mµ such thatfor all x ∈ X and all 0 < r ≤ diam Ω,

M−1r

Q ≤ µ[B(x; r)] ≤M rQ.

The positive real number Q will then be the Hausdorff dimension of (X, d),and the Q-dimensional Hausdorff measure HQ on X will also satisfy the aboveinequalities (possibly with a change in the constant M). A metric space (X, d)is Ahlfors Q-regular if (X, d,HQ) is Ahlfors Q-regular. We use the adjectivesupper or lower to indicate that only one of these inequalities is in force, and—inthe abstract domain setting—add the adjective locally to mean that the requiredinequality holds (or, inequalities hold) for Whitney balls (i.e., for radii 0 < r ≤λd(x)).

There is an interesting result which gives upper estimates for the Assouaddimension of subsets of Ahlfors regular spaces. See [BHR01, 3.12], [DS97, 5.8],[Luu98, 5.2].

2.20. Fact. Suppose X is an Ahlfors Q-regular space and let M ⊂ X. Then

dimAM < Q if and only if M is porous in X; the constants depend only on each

other and the HQ-regularity constant.

The notion of a Loewner space was introduced by Heinonen and Koskela intheir study [HK98] of quasiconformal mappings of metric spaces; Heinonen’srecent monograph [Hei01] renders an enlightening account of these ideas. A


path-connected metric measure space (X, d, µ) is a Q-Loewner space, Q > 1,provided the Loewner control function

ϕ(t) := infmodQ(E,F ;X) : ∆(E,F ) ≤ tis strictly positive for all t > 0; here E, F are non-degenerate disjoint continuain Ω and

∆(E,F ) := dist(E,F )/mindiam(E), diam(F )is the relative size of the pair E, F . Note that we always have Q ≥ dimH(Ω) ≥ 1.

When (Ω, d, µ) is an n-Loewner space with Ω ⊂ Rn a domain and d, µ are

Euclidean distance and Lebesgue n-measure respectively, we simply call Ω aLoewner domain. This is a generalization of Vaisala’s notion of a broad domain

(which he introduced in his analysis [Vai89, 2.15] of space domains QC equivalentto a ball, and also used in his study [NV91, 3.8] of John disks), which in turnis an analog of the quasiextremal distance domains first studied by Gehring andMartio [GM85].

We call Ω ( Rn a ψ-QED domain if ψ : [0,∞) → [0,∞) is a homeomorphism

and for all disjoint continua E, F in Ω,

mod(E,F ; Ω) ≥ ψ(mod(E,F ; Rn)).

Clearly, ψ(t) ≤ t is a necessary restriction on such ψ. Also, every ψ-QED domainis Loewner. The typical nonlinear functions ψ that arise in the literature have theform ψp,M(t) = M

−1 mintp, t1/p with p,M ≥ 1, a condition we call M -QEDp,or simply M -QED when p = 1.

The most important, and original, inequalities of this form are the M-QED

conditions corresponding to ψ(t) = t/M for some constant M ≥ 1. Thisidea was introduced by Gehring and Martio who called such regions quasiex-

tremal distance domains. The terminology arises from the fact that the quantitymod(E,F ; Ω)1/(1−n) is the extremal distance between E and F in Ω. When wespeak of a QED domain or a QED condition, we always mean anM -QED domainor an M -QED condition for some M ≥ 1.

As in [HK96] we can consider the location of the continua E, F as well aslooking at special types of continua. In particular we can relax the ψ-QEDinequality by requiring it to hold only for all disjoint closed balls (or just closedWhitney balls) to get the class ψ-QEDb (or ψ-QEDwb, respectively). Precisedefinitions can be found in [BH06].

Every a-uniform domain in Rn is M -QED for some M = M(a, n); this follows

easily from Jones’ extension result for Sobolev spaces [Jon81, Theorem 1]. Also,it is trivially true that

QED =⇒ ψ − QED =⇒ ψ − QEDb

=⇒ ψ − QEDwb.

The converse of the middle implication fails; see[HK96, Example 4.1] and [BH06,Example 4.2]. According to [BH06, Theorem 3.3], the last implication is re-versible modulo a quantitative change in ψ. In addition, we always have

ψ − QED ⇐⇒ Loewner =⇒ QEDwb

=⇒ ψ − QEDwb

⇐⇒ Loewnerwb


where the last condition means that the Loewner condition is assumed onlyfor Whitney balls. Examples 4.2 and 4.3 in [BH06] illustrate that in generalthe converses of the middle two implications fail to hold. The first equivalenceis established in [BH06, Theorem 1.3]. It remains open as to whether or notLoewner domains (i.e. ψ-QED domains) are always QED.

2.K. Slice Conditions. There are various so-called slice conditions each de-signed to handle their own specific problem. The ideas here are due to Buckleyet al. and his exposition [Buc03] is the place to begin reading about this topic.He and his many co-authors have utilized an assortment of slice conditions toinvestigate all kinds of different problems.

A non-empty bounded open set S ⊂ X is called a C-slice separating x, y

provided

∀ α ∈ Γ(x, y) : ℓ(α ∩ S) ≥ diam(S)/C

and

C−1B(x) ∩ S = ∅ = S ∩ C−1

B(y) .

A set of C-slices for x, y ∈ X is a collection S of pairwise disjoint C-slicesseparating x, y in X. One can show (see [BS03, (2.1)]) that the cardinality ofany such set S of C-slices separating x, y is always bounded by #S ≤ C

2k(x, y).

We are interested in knowing when we can reverse this inequality. Since theremay be no C-slices separating x, y, we consider the quantity

dws(x, y) = dws(x, y;C) = dXws

(x, y;C) := 1 + sup #Swhere the supremum is taken over all S which are sets of C-slices in X separatingx, y, and #S denotes the cardinality of S.

We call (X, d) a weak C-slice space provided for all x, y ∈ X,

k(x, y) ≤ C dws(x, y;C),

Thus in these spaces dws(x, y) ≃ k(x, y), at least when k(x, y) ≥ 2. The weakslice condition was introduced in [BO99, Section 5]; see also [BS03], [Buc03],[Buc04]. When the weak C-slice space (X, d) is a domain Ω ( R

n, we call Ω aweak C-slice domain.

The following rather technical lemma is quite useful for obtaining an up-per bound for the cardinality of a set of slices; in weak slice spaces it providesan upper bound for quasihyperbolic distances. It is the case α = 0 of [BS03,Lemma 2.17].

2.21. Lemma. Let Γ be a 1-rectifiable subset of a rectifiably connected metric

space (X, d). Suppose ϕ : Γ → [ε,∞) (with ε > 0) and S is a collection of

disjoint non-empty bounded subsets of X. Suppose also that there exist positive

constants b, c such that

(a) ∀S ∈ S : ℓ(S ∩ Γ) ≥ c diam(S) ,

(b) ∀S ∈ S , ∀z ∈ S ∩ Γ : ϕ(z) ≤ diam(S) ,


(c) ∀t > 0 : ℓ(ϕ−1(0, t]) ≤ b t .

Then the cardinality of S is at most #S ≤ 2(b/c) log2(4ℓ(Γ)/cε).

3. Uniform Spaces

Roughly speaking, a space is uniform provided points in it can be joined by so-called bounded turning twisted double cone arcs, i.e. paths which are not too longand which stay away from the regions boundary. Uniform domains in Euclideanspace were first studied by John [Joh61] and Martio and Sarvas [MS79] whoproved injectivity and approximation results for them. They are well recognizedas being the ‘nice’ domains for quasiconformal function theory as well as manyother areas of geometric analysis (e.g., potential theory); see [Geh87] and [Vai88].Every (bounded) Lipschitz domain is uniform, but generic uniform domains mayvery well have fractal boundary. Recently, uniform subdomains of the Heisenberggroups, as well as more general Carnot groups, have become a focus of study;see [CT95], [CGN00], [Gre01].

3.A. Euclidean Setting. When our uniform space (see the definition givenbelow in §3.B) (Ω, d) is a domain Ω ⊂ R

n with Euclidean distance, we simplycall Ω a uniform domain. Every plane uniform domain is a quasicircle domain(each of its boundary components is either a point or a quasicircle), and a finitelyconnected plane domain is uniform if and only if it is a quasicircle domain. How-ever, the plane punctured at the integers is not uniform. Such nice topologicalinformation is not true for uniform domains in higher dimensions. For example,a ball with a radius removed is uniform; this is not true when n = 2.

For domains in Rn we can consider uniformity both with respect to the Eu-

clidean distance and with respect to the induced length metric also. The latterclass of domains are usually called inner uniform; cf. [Vai98]. For example, a slitdisk in the plane is not uniform (with respect to Euclidean distance) but it is aninner uniform domain. On the other hand, an infinite strip, or the inside of aninfinite cylinder in space, is not uniform nor inner uniform. The region betweentwo parallel planes is not uniform nor inner uniform. Every quasiball is uniform.

3.B. Measure Metric Space Setting. Following [BHK01], a uniform space isan abstract domain (so, a non-complete, locally complete, rectifiably connectedmetric space) (Ω, d) with the property that there is some constant a ≥ 1 suchthat each pair of points can be joined by an a-uniform arc. A rectifiable arc γjoining x, y in Ω is an a-uniform arc provided

ℓ(γ) ≤ a|x− y|and

minℓ(γ[x, z]), ℓ(γ[y, z]) ≤ a d(z) for all z ∈ γ.

Here ℓ(γ) is the arclength of γ and γ[x, z] denotes the subarc of γ between x, z.The second inequality above ensures that Ω contains the twisted double cone∪B(z; ℓ(z)/a) : z ∈ γ where ℓ(z) denotes the left-hand-side of this inequality;


the first inequality asserts that this twisted double cone is not too ‘crooked’.Consequently, we call γ a double a-cone arc if it satisfies the second inequalityabove (the phrases cigar arc and corkscrew are also used).

3.C. Basic Information. An important, and characteristic, property of uni-form spaces is that quasihyperbolic geodesics are uniform arcs. (See [GO79,Theorems 1,2] for domains in Euclidean space and [BHK01, Theorem 2.10] forgeneral metric spaces.) Slight alterations to the proof of [BHK01, Theorem 2.10]yield the following generalization of this property.

3.1. Fact. In an a-uniform space, quasihyperbolic c-quasigeodesics are b-uniform

arcs where b = b(a, c).

In general, quasihyperbolic geodesics may not exist; see [Vai99, 3.5] for anexample due to P. Alestalo. However, one can still show that quasihyperbolicallyshort arcs are uniform arcs. One can prove that boundary points in a locallycompact uniform space can be joined by quasihyperbolic geodesics, and thesegeodesics are still uniform arcs.

Another crucial piece of information is a characterization of uniformity dueto Gehring and Osgood [GO79, Theorems 1,2]; Bonk, Heinonen, and Koskela[BHK01, Lemma 2.13] verified the necessity of this condition for the metric spacesetting, while the Gehring-Osgood argument can be modified to establish thesufficiency. Recalling the basic estimates for quasihyperbolic distance, we see thatuniform spaces are precisely those abstract domains in which the quasihyperbolicdistance is bilipschitz equivalent to the j distance. See also Theorem 6.1.

3.2. Fact. An abstract domain is a-uniform if and only if k(x, y) ≤ b j(x, y) for

all points x, y. The constants a and b depend only on each other.

We conclude this subsection with a useful fact regarding bounded uniformspaces; see [Her06, Lemmas 2.12,2.13].

3.3. Lemma. Let ρ be a Harnack Koebe density on a bounded a-uniform space

(Ω, d). Suppose that Ωρ is bounded and that quasihyperbolic geodesics in Ω are

double a-cone arcs in Ωρ. Then for any positive constant C,

diamρ[CB(z)] ≃ dρ(z) for all z ∈ Ω,

where the constant depends only on C,Hρ, Kρ, a, λ and the quantity q given in

the proof.

4. Gromov Hyperbolicity

Good sources for information concerning Gromov hyperbolicity include [BHK01],[BBI01], [BS00], [BH99], [Bon96] and especially the references mentioned in theseworks. Vaisala has an especially nice treatment [Vai05a] of Gromov hyperbolicityfor spaces which are not assumed to be geodesic nor proper. Note however thatBonk and Schramm have demonstrated that every Gromov δ-hyperbolic metric


space can be isometrically embedded into some complete geodesic δ-hyperbolicspace; see [BS00, Theorem 4.1].

Hasto [Has06] has an intriguing result giving a striking contrast between thehyperbolicity of (Ω, j) versus that of (Ω, j) for Ω ( R

n: the latter space isalways Gromov hyperbolic whereas the former is Gromov hyperbolic preciselywhen Ω has exactly one boundary point. This is quite surprising as these spacesare bilipschitz equivalent (indeed, j ≤ j ≤ 2j). It is known that for intrinsicspaces, so also for geodesic spaces, Gromov hyperbolicity is preserved under(L,C)-quasiisometries. In particular, Hasto’s result illustrates the failure of thisproperty in the non-intrinsic setting.

4.A. Thin Triangles Definition. A geodesic metric space is Gromov hyper-

bolic if its geodesic triangles are δ-thin for some δ > 0, which means that eachpoint on the edge of any geodesic triangle is within distance δ of some point onone of the other two edges. That is, if [x, y]∪ [y, z]∪ [z, x] is a geodesic triangle,then for all u ∈ [x, y], dist(u, [x, z] ∪ [y, z]) ≤ δ. (Recall that [x, y] denotes somearbitrary, but fixed, geodesic joining x, y.)

There is a more general definition which applies to non-geodesic spaces. It isbased on the Gromov product

(x|y)w :=1

2(|x− w| + |y − w| − |x− y|) for points x, y, w in the space .

The Gromov product is useful even in geodesic spaces; it can be extended to theGromov boundary and then used to define a canonical conformal gauge there.

Roughly speaking, all simply connected manifolds with negative curvature areGromov hyperbolic; e.g., every CAT(κ) space with κ < 0. For a specific example,consider any bounded strictly pseudoconvex domain Ω (with sufficiently smoothboundary) in complex n-space together with any of the classical hyperbolic dis-tances h; a result of Balogh and Bonk [BB00] asserts that (Ω, h) is a Gromovhyperbolic space (with ∂GΩ = ∂Ω, the Euclidean boundary, and canonical con-formal gauge determined by the Carnot-Caratheodory distance on ∂Ω).

4.B. Gromov Boundary. The Gromov boundary ∂GH of a proper geodesicGromov hyperbolic metric space (H, h) is defined as the set of equivalence classesof geodesic rays, with two such rays being equivalent if they have finite Hausdorffdistance. That is, ∂GH is the geodesic boundary of H; see §2.D. An alterna-tive description can be given in terms of (equivalent) sequences which converge at

infinity ; in particular, this allows us to extend the Gromov product to the bound-ary (cf. [Vai05a, 5.7] or [BH99, pp. 431-436]). This in turn yields a canonicallydefined conformal gauge on the Gromov boundary generated by the quasimetrics

qw,ε(ξ, η) = exp[−ε(ξ|η)w] for points ξ, η ∈ ∂GH.

For 0 < ε < ε(δ) = log 2/(4δ), each quasimetric qε is bilipschitz equivalent to anhonest metric on the boundary, and all these metric spaces are QS equivalent toeach other; in particular they all generate the same topology on ∂GH, and ∂GH

is compact. See Fact 2.1, [BH99, Proposition 3.21, pp.435-436], [BHK01, p.18].


4.C. Connection with Uniform Spaces. Bonk, Heinonen and Koskela estab-lished the following fundamental connection between uniform spaces and Gromovhyperbolicity; see [BHK01, Proposition 2.8, Theorem 3.6]

4.1. Fact. The quasihyperbolization (Ω, k) of a locally compact a-uniform space

(Ω, d) is proper, geodesic and δ-hyperbolic where δ = δ(a) = 10000a8. When

(Ω, d) is bounded, (Ω, k) is roughly κ-starlike with κ = 5000a8.

In fact they also prove that the Gromov boundary ∂GΩ ‘is’ the boundary ∂Ω ofΩ in the one-point extension Ω of Ω; see [BHK01, Proposition 3.12]. Moreover, inthe bounded case, the canonical gauge on ∂GΩ is naturally quasisymmetricallyequivalent to the conformal gauge determined by d on ∂Ω. See [Vai05b] forsimilar results in the Banach space setting.

5. Uniformization

The celebrated Riemann Mapping Theorem asserts that every simply con-nected proper subdomain of the plane can be mapped conformally onto theunit disk, and hence supports a bounded conformal uniformizing metric-density,namely, ρ = |f ′| where f is the Riemann map. Koebe proved a similar result forfinitely connected plane domains: any one of these can always be conformallymapped onto a circle domain (meaning that each boundary component is eithera point or a circle).

In space, every conformal map is (the restriction of) a Mobius transformation,and thus the only space regions conformally equivalent to a ball are balls andhalf-spaces. The problem of determining which space domains are QC equivalentto a ball has been investigated for more than four decades by now (see [GV65]),and the most significant result (that I know of) is Vaisala’s characterization in[Vai89] describing the cylindrical domains (Ω = D × R ⊂ R

3) which are QCequivalent to B

3.

5.A. Uniformization Problem. Here we consider the metric space analog ofthe Riemann Mapping Problem. We seek to characterize the abstract domainswhich can be quasiconformally deformed into a uniform space. We ask the ques-tion: What are necessary and sufficient conditions for an abstract domain tosupport a conformal uniformizing metric-density? Theorem A provides an initialanswer.

5.B. BHK Uniformization. Bonk, Heinonen and Koskela developed a uni-formization theory, which they call dampening , valid for proper geodesic Gromovhyperbolic spaces. Their theory produces the following; see [BHK01, Proposition4.5, Chapter 5]. (They established far more than we mention here:-)

5.1. Fact. Let (H, h) be an unbounded proper geodesic Gromov δ-hyperbolic

space. Fix a base point w ∈ H. For ε > 0 define ρε(x) = exp[−εh(x,w)] and let

Hε = (H, dε) where dε = hρε. Then:


(a) The geodesics in H are double a-cone arcs in Hε with a = a(ε, δ) = e1+8εδ.

(b) There is a constant ε0 = ε0(δ) such that for all ε ∈ (0, ε0],

∀x, y ∈ H ,∀ geodesics [x, y] : ℓε[x, y] ≤ 20 dε(x, y);

here ℓε = ℓρε.

In fact, Hε is always bounded, and thus when ε ≤ ε0 we see that (H, h) hasbeen deformed, or dampened, (via the natural metric-density ρε) to a bounded20-uniform space Hε.

We briefly describe their theory in the special case which concerns us.

We consider a locally compact abstract domain (Ω, d) with the property thatthe identity map (Ω, ℓ) → (Ω, d) is a homeomorphism (so the identity (Ω, k) →(Ω, d) is also a homeomorphism) and such that its quasihyperbolization (Ω, k)is a Gromov hyperbolic space. According to Fact 5.1, the space (Ω, k) admits auniformizing density of the form

ρε(x) := exp[−εk(x,w)];

here w ∈ Ω is a fixed base point and ε > 0 a sufficiently small parameter. Moreprecisely, when (Ω, k) is δ-hyperbolic and 0 < ε < ε(δ), the quasihyperbolicgeodesics in Ω are 20-uniform arcs in (Ω, dε). (A careful check of BHK showsthat ε(δ) = [42(5 + 192δ + 1920δ2)]−1 ≤ (300 max1, δ)−2. :-) Here dε standsfor the distance function obtained by conformally deforming k via the metric-density ρε. Since k was obtained from the original distance function d via thequasihyperbolic density 1/d, Ωε = (Ω, dε) is a conformal deformation of (Ω, d)via the metric-density

πε(x) := ρε(x)/d(x) = d(x)−1 exp(−εk(x,w)) ;

again, πε will be a uniformizing density when 0 < ε < ε(δ) = (300 max1, δ)−2.

In order to determine when πε will be a Harnack or Koebe density, we needthe following information concerning ρε (see [BHK01, (4.4),(4.6),(4.17)]):

e−εk(x,y) ≤ ρε(x)

ρε(y)≤ e

εk(x,y),(5.2)

ρε(x)

eε≤ dε(x) ≤ (2eεκ − 1)

ρε(x)

ε.(5.3)

The first set of inequalities (5.2) hold for all points x, y ∈ Ω and all ε > 0. Theyguarantee that the identity map (Ω, k) → (Ω, dε) is locally bilipschitz, so (Ω, dε)is locally compact and rectifiably connected. On the other hand, Ωε = (Ω, dε) isnon-complete, so we can form Ωε, put ∂εΩ = Ωε \Ω and let dε(x) = distε(x, ∂εΩ).(See [BHK01, pp.27-28]). We find that the leftmost inequality in (5.3) holds forall x ∈ Ω and any ε > 0. However, in order to obtain the rightmost inequalityin (5.3), we must further require that (Ω, k) be roughly κ-starlike.

Now using (5.2), and obvious inequalities for 1/d, we deduce that

e−2R ≤ e

−(ε+1)R ≤ πε(y)

πε(x)≤ e

(ε+1)R ≤ e2R


for all points x, y with k(x, y) ≤ R. According to Lemma 2.12, when (Ω, d) islocally a-quasiconvex we have

τB(x) ⊂ Bk(x;R) provided 0 < τ ≤ minλ,R/[a(1 +R)].Taking R = a (say) and τ = minλ, 1/2a we find that for all x ∈ Ω and ally ∈ τB(x),

e−2a ≤ e

−(ε+1)a ≤ πε(y)

πε(x)≤ e

(ε+1)a ≤ e2a;

that is, πε is a Harnack density with constants H = e2a and τ which are inde-

pendent of ε.

Finally, since Ωε is non-complete, we can ask whether or not πε is a Koebedensity. Since πε(x) = ρε(x)/d(x), we see from (5.3) that πε will be a Koebedensity, with constant K = (2eεκ − 1)/ε (assuming εe ≤ 1), provided (Ω, k) isroughly κ-starlike (and (Ω, d) uniformly locally quasiconvex).

These conditions describing when πε will be a Harnack or Koebe density donot require that (Ω, k) be Gromov hyperbolic. We record the above informationfor later reference; see also Lemma 2.8 and Corollary 5.8. Note too that (Ω, dε)is bounded with diamε Ωε ≤ 2/ε.

5.4. Lemma. Let (Ω, d) be a locally a-quasiconvex abstract domain and fix a

base point w ∈ Ω. Then for any ε > 0,

πε(x) = ρε(x)/d(x) = d(x)−1 exp(−εk(x,w))

is a Harnack density with constant H = e2a. If in addition (Ω, k) is roughly κ-

starlike, then πε is also a Koebe density, with constant K = maxεe, (2eεκ−1)/ε.The above, in conjunction with Lemma 2.11 and Proposition 5.6(b), provides

a one-to-one correspondence between conformal metric-densities on Ω and thesame on Ωε. Here is a precise statement of this.

5.5. Corollary. Suppose (Ω, d, µ) is an abstract domain having bounded geom-

etry and a Gromov hyperbolic roughly starlike quasihyperbolization (Ω, k). Let

Ωε = (Ω, dε, µε) be the deformation of Ω via the density πε defined just above.

If σ is a conformal density on Ωε, then its pull-back ρ = σ πε is a conformal

density on Ω, and conversely if ρ is a conformal density on Ω, then its push-

forward σ = ρ π−1

ε is a conformal density on Ωε. In both cases Ωρ = (Ωε)σ, and

the metric-density parameters depend only on each other and the data associated

with Ω.

5.C. Bounded Geometry and its Consequences. Recall our definition thatan abstract domain (Ω, d, µ) is a locally complete, rectifiably connected, non-complete metric measure space; these are our standing metric hypotheses. Wesay that (Ω, d, µ) has bounded Q-geometry , Q > 1, provided it is both locally

upper Ahlfors Q-regular (see §2.J) and weakly locally Q-Loewner ; this lattercondition means that there exists a positive constant m such that for all x ∈ Ω,and all non-degenerate disjoint continua E, F in λB(x),

∆(E,F ) ≤ 16 =⇒ mod Q(E,F ; Ω) ≥ m.


Any space Ω with the property that Whitney balls λB(x) (with a fixed parame-ter) are uniformly bilipschitz equivalent to Euclidean balls (in a fixed dimension)is easily seen to have bounded geometry. Other examples include Riemann-ian manifolds with Ricci bounded geometry as well as the exotic examples ofBourdon-Pajot and Laakso for any Q > 1; see [BHK01, Exs.9.7, p.86] and thereferences mentioned there. (Note that by Proposition 5.6, (Ω, d, µ) has boundedgeometry if and only if its quasihyperbolization satisfies the condition studied in[BHK01, Chpt.9].)

The first part of bounded Q-geometry is a necessary condition for Ω to supportany conformal density (at least when Ω is locally quasiconvex). The second partof bounded Q-geometry, the weak local Loewner criterion, can also be describedin terms of Poincare inequalities as explained in [BHK01, Proposition 9.4] and[HK98, §5]; it ensures that there are plenty of curves available (e.g., it gives localquasiconvexity). However, to substantiate this existence of many curves requiresthe use of certain modulus estimates (see Facts 2.15,2.16), and these estimatesin turn require an upper mass condition.

The Loewner part of bounded Q-geometry also performs an essential role intwo other places. First, it is a key player in the proof of Proposition 2.9, whichis the crucial ingredient in the proof of (d) implies (e) in Theorem D. Second,for uniform Loewner spaces with appropriately ‘thick’ boundaries, the Koebecondition for a metric-density follows from the Harnack and Ahlfors condition;this fact is utilized in the proof of Theorem C.

Bounded geometry provides a number of essential properties for our underlyingspace.

5.6. Proposition. Let (Ω, d, µ) be an abstract domain having bounded Q-

geometry (with constants M,m, λ). Then:

(a) µ is locally Ahlfors Q-regular.

(b) Ω is locally quasiconvex with constants which depend only on Q,M,m, λ.

(c) Ω is locally Q-Loewner with a control function ψ and parameters κ, ε0 which

depend only on the data Q,M,m, λ associated with Ω.

Here is a useful consequence of the above.

5.7. Corollary. Let ρ be a Harnack Koebe density on an abstract domain

(Ω, d, µ) having bounded Q-geometry. Then Ωρ is locally Ahlfors Q-regular and

locally Q-Loewner with parameters and a control function which depend only on

the data associated with ρ and Ω.

Note that if Ωρ above is also uniform, then it would be globally Loewner by[BHK01, Theorem 6.4]. We record the following consequence of this observation.

5.8. Corollary. Let (Ω, d, µ) be an abstract domain having bounded Q-geometry

(with constants M,m) and a Gromov δ-hyperbolic roughly κ-starlike quasihy-

perbolization (Ω, k). Then any BHK-uniformization Ωε = (Ω, dε, µε) of Ω is a


bounded locally Ahlfors Q-regular locally Q-Loewner space (hence of bounded Q-

geometry), and even Q-Loewner when ε < ε(δ). Here the various parameters and

control functions depend only on δ, κ,Q,M,m, λ, ε.

5.D. Lifts and Metric Doubling Measures. Here we discuss the ideas be-hind Theorem C and briefly outline the proof. Recall the notion of a metricdoubling measure discussed near the end of §2.F.

We define the lift ρν of ν via the formula

(5.9) ρν(x) :=ν(Σx)

1/P

d(x);

here ν is a P -dimensional metric doubling measure on ∂GΩ and, for some fixedbase point w ∈ Ω, the shadow of a point x ∈ Ω is

Σx := ζ ∈ ∂GΩ : k(x, [w, ζ)) ≤ R.It is not hard to see that there are constants R = R(δ, κ, ε) and C = C(δ, κ, ε)such that

(5.10) Sx := ∂εΩ ∩ 2Bε(x) ⊂ Σx ⊂ ∂εΩ ∩ CBε(x);

of course we are using the natural identification of ∂GΩ with ∂εΩ. Employing(5.10), the doubling property of ν, and the fact that πε is Koebe, it is straight-forward to verify that the push-forward of ρν (as defined in (5.9)) via the uni-formizing density πε gives a density on Ωε which is bilipschitz equivalent to thedensity defined via the formula

ρ(x) :=ν(Sx)

1/P

dε(x)where Sx := ∂εΩ ∩ 2Bε(x).

Theorem C now follows from Corollary 5.5 once we verify that ρ is a BorelHarnack Ahlfors Koebe doubling metric-density on Ωε. () The first two ofthese are easy. The Koebe property follows from Theorem E once we know theAhlfors volume growth property. To see that Theorem E can be applied, wefirst use Fact 2.2 along with Theorem F to see that the Whitney ball modulusproperty holds.

To establish the Ahlfors property we need the doubling property. This in turnrequires the following ‘quasihyperbolic doubling’ result.

5.11. Proposition. Let (Ω, d, µ) be a locally a-quasiconvex locally Ahlfors Q-

regular abstract domain. Then its quasihyperbolization (Ω, k) is locally doubling

in the sense that if Σ ⊂ Ω is a set of points satisfying

0 < t ≤ k(x, y) ≤ T <∞ for all x, y ∈ Σ, x 6= y,

then the cardinality of Σ is bounded by

#Σ ≤ 2M28Q(2M224Q)8aT/λt.

Here M is the local regularity constant and λ the Whitney ball constant.


Here is an interesting application of Theorems C and D which provides a char-acterization of doubling conformal densities in terms of metric doubling measures.We say that a metric-density ρ on Ω is induced by a metric doubling measure ν

on ∂GΩ if there exists a constant C ≥ 1 such that

C−1ρν(x) ≤ ρ(x) ≤ Cρν(x) for µ-a.e. x ∈ Ω.

5.12. Theorem. Assume the basic minimal hypotheses, that the Gromov bound-

ary of Ω is uniformly perfect, and P < Q. A metric-density ρ on Ω is a doubling

conformal density, with (∂ρΩ, dρ) Ahlfors P -regular, if and only if ρ is induced

by some P -dimensional metric doubling measure on ∂GΩ.

5.E. Volume Growth Problem. Here we briefly outline the proof of Theo-rem A. Recall that this result provides our answer to the problem of decidingwhen there exists a uniformizing conformal density (so, in particular, the asso-ciated measure (2.7) should have Ahlfors regular volume growth). The necessityin this result follows from Fact 4.1 along with Fact 2.20. The real work involvedis in establishing the sufficiency.

The first major step is to prove Theorem D. Following [BHR01], we say thata metric-density ρ, on a uniform space (Ω, d, µ), is doubling provided µρ is adoubling measure on (Ω, d); i.e., there exists a constant D = Dρ such that for allx ∈ Ω,

(D) µρ[B(x; 2r)] ≤ Dµρ[B(x; r)] for all r > 0.

When Ω is not uniform, the above doubling condition may fail to hold even if ρis ‘nice’; e.g., consider ρ = |f ′| on Ω = x + iy : |y| < 1 in the complex plane,where f is a conformal homeomorphism of Ω onto the unit disk. To compensatefor this we employ the following definition: a conformal density ρ, on an abstractdomain (Ω, d, µ) with Gromov hyperbolic quasihyperbolization, is doubling if itspush-forward is doubling on some BHK-uniformization Ωε of Ω. (Since all suchspaces Ωε are QS equivalent, and doubling measures can be defined for conformalgauges, there is no ambiguity here. See §2.C for a discussion of conformal gauges,and also the very last paragraph of §2.F.)

Notice that the doubling condition (D) uses d-balls but µρ-measure and thusinterweaves the measure properties of the deformed space with the metric prop-erties of the original (or uniformized) space.

Our proof of Theorem D requires the Gehring-Hayman Inequality (Theo-rem B), Corollary 2.10, and the following two results. First we give a necessarycondition for a metric-density to be doubling.

5.13. Proposition. Let ρ be a Harnack density on an a-uniform locally Ahlfors

Q-regular space (Ω, d, µ) (with constants H,M, λ). Suppose that ρ is also doubling

on Ω (with constant D). Then quasihyperbolic geodesics in Ω are double c-cone

arcs in Ωρ where c = c(D,H,M,Q, a, λ). If in addition Ω is bounded, then so is

Ωρ and ∂Ω ⊂ ∂ρΩ, with equality holding when Ω is also locally Q-Loewner.

Next we state a sufficient condition for a metric-density to be doubling.


5.14. Proposition. Let ρ be a conformal density on an a-uniform locally

lower Ahlfors Q-regular space (Ω, d, µ) (with constants H,A,K,M, λ). Sup-

pose that quasihyperbolic geodesics in Ω are double c-cone arcs in Ωρ. Then

Ωρ is Ahlfors Q-regular with a parameter which depends only on c and the data

H,A,K,M,Q, a, λ associated with ρ and Ω. If in addition Ω and Ωρ are both

bounded, then ρ is doubling on Ω with a parameter which depends only on the

aforementioned data and the quantity q given in the proof of Lemma 3.3.

The second major step in the proof of Theorem A is to establish Theorem C.Its proof is outlined above in §5.D. That done, we use Theorem C to obtain adoubling conformal density, which by Theorem D is also bounded and uniformiz-ing.

6. Characterizations of Uniform Spaces

Here we mention a number of characterizations for uniform spaces and uni-form domains. In [Vai88] Vaisala provides a complete description of the variouspossible twisted double cone conditions (which he calls length cigars, diameter

cigars, distance cigars, and Mobius cigars). The work [Mar80] of Martio shouldalso be mentioned.

6.A. Metric Characterizations. We have already mentioned that uniformspaces are precisely the abstract domains in which the quasihyperbolic metric isbilipschitz equivalent to the so-called j metric; see Fact 3.2.

It turns out that the following seemingly weaker quasihyperbolic metric con-dition also characterizes uniform spaces. For uniform subdomains of Banachspaces this result is due to Vaisala [Vai91, 6.16, 6.17]. Here we write

r(x, y) :=|x− y|

d(x) ∧ d(y)to denote the so-called relative distance between x, y. See [BH07] for the followingversion.

6.1. Theorem. A locally quasiconvex abstract domain is uniform if and only if

there is a homeomorphism ϑ : [0,∞) → [0,∞) satisfying lim supt→∞ ϑ(t)/t < 1,and such that for all points x, y, k(x, y) ≤ ϑ (r(x, y)). The uniformity constant

depends only on ϑ, and conversely in an a-uniform space, one can always take

ϑ(t) = b log(1 + t) with b = b(a).

6.B. Gromov Boundary Characterizations. We call Ω ( Rn a Gromov

domain if its quasihyperbolization (Ω, k) is Gromov hyperbolic. Bonk, Heinonenand Koskela corroborated the following [BHK01, Proposition 7.12].

6.2. Fact. A Gromov domain in Euclidean space is uniform if and only if it is

linearly locally connected.

They utilized the above to establish the following [BHK01, Theorem 7.11].


6.3. Fact. A (bounded) Gromov domain in Euclidean space is uniform if and

only if the canonical gauge on the Gromov boundary is quasisymmetrically equiv-

alent to the Euclidean gauge on the Euclidean boundary.

The careful reader will recognize that the Bonk-Heinonen-Koskela results wereestablished for regions on the sphere (i.e., using the spherical metric). Resultsof Balogh and Buckley [BB06] are useful in this regard.

Vaisala has recently proven a Banach space analog of the above result; see[Vai05b]. Along with providing a dimension free version of this result, he alsoconsiders arbitrary domains (not just bounded) and replaces QS equivalence withQM equivalence. In addition, he provides an example of a Gromov hyperbolicdomain which is LLC but not uniform.

6.C. Characterizations using QC Maps. It is evident that bilipschitz ho-meomorphisms map uniform spaces to uniform spaces. This also holds true forQS and QM maps of uniform domains in Euclidean space and in Banach spaces[Vai99, Theorem 10.22], but not in the general metric space setting, and not forQC maps. On the other hand, according to [BKR98, 2.4], the average derivativeof a quasiconformal map f : B

n → Ω ⊂ Rn is a conformal metric density on B

n

(a uniform n-Loewner n-regular space). Thus we can appeal to Theorem D andread off a number of conditions which characterize when Ω will be uniform.

6.D. Capacity Conditions. It is known that given 0 < λ ≤ 1/2, there existsa constant c = c(λ, n) > 0 such that

∀ x, y ∈ Ω : k(x, y) ≥ 2 =⇒ mod(λB(x), λB(y);D) ≥ c/k(x, y)n−1;

this is valid for any proper subdomain Ω of Rn. To prove it, one starts by us-

ing Lemma 2.14 to select an appropriate cover of any quasihyperbolic geodesicjoining x, y, and then a standard application of the Poincare inequality appliedto adjacent balls leads to the asserted inequality. See the proof of [HK96, Theo-rem 6.1].

Let C > 0 and 0 < λ ≤ 1/2. A proper subdomain Ω of Rn is a (C, λ)-k-cap

domain provided

∀ x, y ∈ D : k(x, y) ≥ 2 =⇒ mod(λB(x), λB(y);D) ≤ C/k(x, y)n−1.

Thus in a k-cap domain D, we have mod(λB(x), λB(y);D) ≃ k(x, y)1−n forpoints with k(x, y) ≥ 2, with constants of comparison dependent only on λ, n,and the k-cap parameter.

This is the two-sided version of a condition introduced by Buckley in [Buc04] tostudy quasiconformal images of domains which satisfy a quasihyperbolic bound-ary condition. As explained on p.26 of that paper, a (C, λ)-k-cap conditionimplies a (C ′

, λ′)-k-cap condition for some C ′ = C

′(C, λ, λ′, n). We mainly con-sider the case λ = 1/2, and refer to a (C, 1/2)-k-cap domain simply as a C-k-cap

domain.


Every uniform domain in Rn is a k-cap domain, and the class of k-cap do-

mains is invariant under quasiconformal mappings (with a quantitative changeof parameter C). For proofs of these statements see [Buc04].

Recently we established the following characterization for uniform domains inEuclidean space; see [BH06, Theorem 3.5].

6.4. Theorem. A proper subdomain of Rn is uniform if and only if it is both

QEDwb and a k-cap domain.

This result is quantitative.

6.E. LLC and Slice Conditions. By utilizing certain slice conditions, Baloghand Buckley [BB03] established a number of geometric characterizations for Gro-mov hyperbolic spaces. Here we mention the following new characterization ofuniform spaces; see [BH07]

6.5. Theorem. An abstract domain is uniform and LEC if and only if it is

quasiconvex, LLC2 with respect to arcs, and satisfies a weak slice condition.

These implications are quantitative.

We have modified the usual LLC conditions (see §2.E) by requiring that thepoints in question be joinable by rectifiable arcs (rather than just by continua).Every uniform domain in R

n is LLC. In fact every uniform space is quasiconvexand thus LLC1 with respect to arcs. However, uniform spaces need not be LLC2;e.g., an ‘asterik’ type space (the disjoint union of a point and a bunch of linesegments or rays, with its intrinsic length distance) may be uniform but notLLC2. We say that a locally complete metric space is locally externally connected,abbreviated LEC, provided there is a constant c ≥ 1 such that the (LLC2)condition holds for all points x ∈ Ω and all r ∈ (0, d(x)/c).

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David A Herron E-mail: [email protected]: Department of Mathematics, University of Cincinnati, OH 45221, USA


p-Laplace operator, quasiregular mappings, andPicard-type theorems

Ilkka Holopainen and Pekka Pankka

Abstract. We describe the role of p-harmonic functions and forms in thetheory of quasiregular mappings.

Keywords. Quasiregular mapping, p-harmonic function, p-harmonic form,conformal capacity.

2000 MSC. Primary 58J60, 30C65; Secondary 53C20, 31C12, 35J60.

Contents

1. Introduction 117

2. A-harmonic functions 120

3. Morphism property and its consequences 123

3.1. Sketch of the proof of Reshetnyak’s theorem 125

Further properties of f 127

4. Modulus and capacity inequalities 128

5. Liouville-type results for A-harmonic functions 130

6. Liouville-type results for quasiregular mappings 136

7. Picard-type theorems 137

8. Quasiregular mappings, p-harmonic forms, and de Rham cohomology 145

References 148

1. Introduction

In this survey we emphasize the importance of the p-Laplace operator as atool to prove basic properties of quasiregular mappings, as well as Liouville-and Picard-type results for quasiregular mappings between given Riemannianmanifolds. Quasiregular mappings were introduced by Reshetnyak in the midsixties in a series of papers; see e.g. [36], [37], and [38]. An interest in studyingthese mappings arises from a question about the existence of a geometric functiontheory in real dimensions n ≥ 3 generalizing the theory of holomorphic functionsC → C.

118 Ilkka Holopainen and Pekka Pankka IWQCMA05

Definition 1.1. A continuous mapping f : U → Rn of a domain U ⊂ R

n iscalled quasiregular (or a mapping of bounded distortion) if

(1) f ∈ W1,nloc

(U ; Rn), and(2) there exists a constant K ≥ 1 such that

|f ′(x)|n ≤ K Jf (x) for a.e. x ∈ U.

The condition (1) means that the coordinate functions of f belong to thelocal Sobolev space W 1,n

loc(U) consisting of locally n-integrable functions whose

distributional (first) partial derivatives are also locally n-integrable. In Condi-tion (2) f ′(x) denotes the formal derivative of f at x, i.e. the n × n matrix(Djfi(x)

)defined by the partial derivatives of the coordinate functions fi of f .

Furthermore,

|f ′(x)| = max|h|=1

|f ′(x)h|

is the operator norm of f ′(x) and Jf (x) = det f ′(x) is the Jacobian determinantof f at x. They exist a.e. by (1). The smallest possible K in Condition (2) isthe outer dilatation KO(f) of f . If f is quasiregular, then

Jf (x) ≤ K′ℓ(f ′(x))n a.e.

for some constant K ′ ≥ 1, where

ℓ(f ′(x)) = min|h|=1

|f ′(x)h|.

The smallest possible K′ is the inner dilatation KI(f) of f . It is easy to

see by linear algebra that KO(f) ≤ KI(f)n−1 and KI(f) ≤ KO(f)n−1. IfmaxKO(f), KI(f) ≤ K, f is called K-quasiregular .

To motivate the above definition, let us consider a holomorphic functionf : U → C, where U ⊂ C is an open set. We write f as a mapping f =(u, v) : U → R

2, U ⊂ R

2,

f(x, y) =(u(x, y), v(x, y)

).

Then u and v are harmonic real-valued functions in U and they satisfy theCauchy-Riemann system of equations

D1u = D2v

D2u = −D1v,

whereD1 = ∂/∂x, D2 = ∂/∂y. For every (x, y) ∈ U , the differential f ′(x, y) : R2 →

R2 is a linear map whose matrix (with respect to the standard basis of the plane)

is (D1u D2u

D1v D2v

)=

(D1u D2u

−D2u D1u

).

Hence

(1.1) |f ′(x, y)|2 = det f ′(x, y).

p-Laplace operator, quasiregular mappings, and Picard-type theorems 119

The first trial definition for mappings f : U → Rn of a domain U ⊂ R

n, sharingsome geometric and topological properties of holomorphic functions, could bemappings satisfying a condition

(1.2) |f ′(x)|n = Jf (x), x ∈ U.

However, it has turned out that, in dimensions n ≥ 3, a mapping f : U → Rn

belonging to the Sobolev space W 1,nloc

(U ; Rn) and satisfying (1.2) for a.e. x ∈ U iseither constant or a restriction of a Mobius map. This is the so-called generalizedLiouville theorem due to Gehring [12] and Reshetnyak [38]; see also the thoroughdiscussion in [29].

Next candidate for the definition is obtained by replacing the equality (1.2)by a weaker condition

(1.3) |f ′(x)|n ≤ K Jf (x) a.e. x ∈ U,

where K ≥ 1 is a constant. Note that Jf (x) ≤ |f ′(x)|n holds for a.e. x ∈ U . Nowthere remains a question on the regularity assumption of such mapping f. Againthere is some rigidity in dimensions n ≥ 3. Indeed, if a mapping f satisfying(1.3) is non-constant and smooth enough (more precisely, if f ∈ C

k, with k = 2

for n ≥ 4 and k = 3 for n = 3), then f is a local homeomorphism. Furthermore,it then follows from a theorem of Zorich that such mapping f : R

n → Rn is

necessarily a homeomorphism, for n ≥ 3; see [47]. We would also like a class ofmaps satisfying (1.3), with fixedK, to be closed under local uniform convergence.In order to obtain a rich enough class of mappings, it is thus necessary to weakenthe regularity assumption from C

k-smoothness. See [15], [5], and [32] for recentdevelopments regarding smoothness and branching of quasiregular mappings.

The basic analytic and topological properties of quasiregular mappings arelisted in the following theorem by Reshetnyak; see [39], [41].

Theorem 1.2 (Reshetnyak’s theorem). Let U ⊂ Rn be a domain and let f : U →

Rn be quasiregular. Then

(1) f is differentiable a.e. and

(2) f is either constant or it is discrete, open, and sense-preserving.

Recall that a map g : X → Y between topological spaces X and Y is discrete

if the preimage g−1(y) of every y ∈ Y is a discrete subset of X and that g is open

if gU is open for every open U ⊂ X. We also remark that a continuous discreteand open map g : X → Y is called a branched covering.

To say that f : U → Rn is sense-preserving means that the local degree

µ(y, f,D) is positive for all domains D ⋐ U and for all y ∈ fD \ f∂D. Thelocal degree is an integer that tells, roughly speaking, how many times f wrapsD around y. It can be defined, for example, by using cohomology groups withcompact support. For the basic properties of the local degree, we refer to [41,Proposition I.4.4]; see also [11], [35], and [45]. For example, if f is differentiableat x0 with Jf (x0) 6= 0, then µ(f(x0), f,D) = sign Jf (x0) for sufficiently small


connected neighborhoods D of x0. Another useful property is the following ho-motopy invariance: If f and g are homotopic via a homotopy ht, h0 = f, h1 = g,

such that y ∈ htD \ ht∂D for every t ∈ [0, 1], then µ(y, f,D) = µ(y, g,D).

y

f

D

fD

f∂D

The definition of quasiregular mappings extends easily to the case of continu-ous mappings f : M → N, where M and N are connected oriented Riemanniann-manifolds.

Definition 1.3. A continuous mapping f : M → N is quasiregular (or a mappingof bounded distortion) if it belongs to the Sobolev space W 1,n

loc(M ;N) and there

exists a constant K ≥ 1 such that

(1.4) |Txf |n ≤ KJf (x) for a.e. x ∈M.

Here again Txf : TxM → Tf(x)N is the formal differential (or the tangentmap) of f at x, |Txf | is the operator norm of Txf , and Jf (x) is the Jacobiandeterminant of f at x uniquely defined by (f ∗volN)x = J(x, f)(volM)x almosteverywhere. Note that Txf can be defined for a.e. x by using partial derivativesof local expressions of f at x. The geometric interpretation of (1.4) is that Txfmaps balls of TxM either to ellipsoids with controlled ratios of the semi-axes orTxf is the constant linear map.

TxM

M N

Txf

f

Tf(x)N

We assume from now on that M and N are connected oriented Riemanniann-manifolds.

2. A-harmonic functions

It is well-known that the composition uf of a holomorphic function f : U → C

and a harmonic function u : fU → R is a harmonic function in U . In other words,holomorphic functions are harmonic morphisms. Quasiregular mappings have asomewhat similar morphism property: If f : U → R

n is quasiregular and u is ann-harmonic function in a neighborhood of fU , then uf is a so-called A-harmonic


function in U . In this section we introduce the notion of A-harmonic functionsand recall some of their basic properties that are relevant for this survey.

We denote by 〈·, ·〉 the Riemannian metric of M . Recall that the gradient ofa smooth function u : M → R is the vector field ∇u such that

〈∇u(x), h〉 = du(x)h

for every x ∈M and h ∈ TxM.

The divergence of a smooth vector field V can be defined as a functiondiv V : M → R satisfying

LV ω = (div V )ω,

where ω = volM is the (Riemannian) volume form and

LV ω = limt→0

(αt)∗ω − ω

t

is the Lie derivative of ω with respect to V, and α is the flow of V.V

x

αt(x)

α = flow of V

We say that a vector field ∇u ∈ L1

loc(M) is a weak gradient of u ∈ L

1

loc(M) if

(2.1)

∫

M

〈∇u, V 〉 = −∫

M

u div V

for all vector fields V ∈ C∞0

(M). Conversely, a function div V ∈ L1

loc(M) is

a weak divergence of a (locally integrable) vector field V if (2.1) holds for allu ∈ C

∞0

(M). Note that∫M

div Y = 0 if Y is a smooth vector field in M withcompact support.

We define the Sobolev space W 1,p(M) and its norm as

W1,p(M) = u ∈ L

p(M) : weak gradient ∇u ∈ Lp(M), 1 ≤ p <∞,

‖u‖1,p = ‖u‖p + ‖|∇u|‖p.

Let G ⊂ M be open. Suppose that for a.e. x ∈ G we are given a continuousmap

Ax : TxM → TxM

such that the map x 7→ Ax(X) is a measurable vector field whenever X is.Suppose that there are constants 1 < p <∞ and 0 < α ≤ β <∞ such that

〈Ax(h), h〉 ≥ α|h|p

and

|Ax(h)| ≤ β|h|p−1


for a.e. x ∈ G and for all h ∈ TxM. In addition, we assume that for a.e. x ∈ G

〈Ax(h) −Ax(k), h− k〉 > 0

whenever h 6= k, and

Ax(λh) = λ|λ|p−2Ax(h)

whenever λ ∈ R \ 0.A function u ∈ W

1,ploc

(G) is called a (weak) solution of the equation

(2.2) − divAx(∇u) = 0

in G if ∫

G

〈Ax(∇u),∇ϕ〉 = 0

for all ϕ ∈ C∞0

(G). Continuous solutions of (2.2) are called A-harmonic functions

(of type p). By the fundamental work of Serrin [43], every solution of (2.2) hasa continuous representative. In the special case Ax(h) = |h|p−2

h, A-harmonicfunctions are called p-harmonic and, in particular, if p = 2, we obtain the usualharmonic functions. The conformally invariant case p = n = the dimension ofM is important in the sequel. In this case p-harmonic functions are called, ofcourse, n-harmonic functions.

A function u ∈ W1,ploc

(G) is a subsolution of (2.2) in G if

− divAx(∇u) ≤ 0

weakly in G, that is ∫

G

〈Ax(∇u),∇ϕ〉 ≤ 0

for all non-negative ϕ ∈ C∞0

(G). A function v is called supersolution of (2.2) if−v is a subsolution. The proofs of the following two basic estimates are straight-forward once the appropriate test function is found. Therefore we just give thetest function and leave the details to readers.

Lemma 2.1 (Caccioppoli inequality). Let u be a positive solution of (2.2) (for

a given fixed p) in G and let v = uq/p, where q ∈ R \ 0, p− 1. Then

(2.3)

∫

G

ηp|∇v|p ≤

(β|q|

α|q − p+ 1|

)p ∫

G

vp|∇η|p

for every non-negative η ∈ C∞0

(G).

Proof. Write κ = q − p+ 1 and use ϕ = uκηp as a test function.

Remark 2.2. In fact, the estimate (2.3) holds for positive supersolutions ifq < p− 1, q 6= 0, and for positive subsolutions if q > p− 1.

The excluded case q = 0 above corresponds to the following logarithmic Cac-cioppoli inequality.


Lemma 2.3 (Logarithmic Caccioppoli inequality). Let u be a positive superso-

lution of (2.2) (for a given fixed p) in G and let C ⊂ G be compact. Then

(2.4)

∫

C

|∇ log u|p ≤ c

∫

G

|∇η|p

for all η ∈ C∞0

(G), with η|C ≥ 1, where c = c(p, β/α).

Proof. Choose ϕ = ηpu

1−p as a test function.

These two lemmas together with the Sobolev and Poincare inequalities areused in proving Harnack’s inequality for non-negative A-harmonic functions bythe familiar Moser iteration scheme. In the following |A| denotes the volume ofa measurable set A ⊂M.

Theorem 2.4 (Harnack’s inequality). Let M be a complete Riemannian mani-

fold and suppose that there are positive constants R0, C, and τ ≥ 1 such that a

volume doubling property

(2.5) |B(x, 2r)| ≤ C |B(x, r)|holds for all x ∈ M and 0 < r ≤ R0, and that M admits a weak (1, p)-Poincare

inequality

(2.6)

∫

B

|v − vB| ≤ C r

∫

τB

|∇v|p

1/p

for all balls B = B(x, r) ⊂ M, with τB = B(x, τr) and 0 < r ≤ R0, and for all

functions v ∈ C∞(B). Then there is a constant c such that

(2.7) supB(x,r)

u ≤ c infB(x,r)

u

whenever u is a non-negative A-harmonic function in a ball B(x, 2r), with 0 <r ≤ R0.

In particular, if the volume doubling condition (2.5) and the Poincare inequal-ity (2.6) hold globally, that is, without any bound on the radius r, we obtain aglobal Harnack inequality. We refer to [18], [9], and [16] for proofs of the Harnackinequality.

3. Morphism property and its consequences

The very first step in developing the theory of quasiregular mappings is toprove, by direct computation, that quasiregular mappings have the morphismproperty in a special case where the n-harmonic function is smooth enough.

Theorem 3.1. Let f : M → N be a quasiregular mapping (with a constant K)

and let u ∈ C2(N) be n-harmonic. Then v = u f is A-harmonic (of type n) in

M, with

(3.1) Ax(h) = 〈Gxh, h〉n

2−1Gxh,


where Gx : TxM → TxM is given by

Gxh =

Jf (x)

2/nTxf

−1(Txf−1)Th, if Jf (x) exists and is positive,

h, otherwise.

The constants α and β for A depend only on n and K.

Proof. Let us first write the proof formally and then discuss the steps in moredetail. In the sequel ω stands for the volume forms in M and N. Let V ∈ C

1(M)be the vector field V = |∇u|n−2∇u. Since u is n-harmonic and C

2-smooth, wehave div V = 0. By Cartan’s formula we obtain

d(V yω) = d(V yω) + V y (dω) = LV ω = (div V )ω = 0

since dω = 0. Here Xy η is the contraction of a differential form η by a vectorfield X. Thus, for instance, V yω is the (n− 1)-form

V yω(·, . . . , ·︸︷︷︸n−1

) = ω(V, ·, . . . , ·︸︷︷︸n−1

).

Hence

(3.2) df∗(V yω) = f

∗d(V yω)

a.e.= 0.

On the other hand, we have a.e. in M

(3.3) f∗(V yω) = Wy f

∗ω = Wy (Jfω) = JfWyω,

where W is a vector field that will be specified later (roughly speaking, f∗W =V ). We obtain

(3.4) d(JfWyω) = 0,

or equivalently

(3.5) div(JfW ) = 0

which can be written as

(3.6) divAx(∇v) = 0,

where A is as in the claim.

Some explanations are in order. When writing

f∗d(V yω)

a.e.= 0,

we mean that for a.e. x ∈ U and for all vectors v1, v2, . . . , vn ∈ TxM

f∗d(V yω)(v1, v2, . . . , vn) = d(V yω)(f∗v1, f∗v2, . . . , f∗vn) = 0,

where f∗ = f∗,x = Txf is the tangent mapping of f at x. The equality on the

left-hand side of (3.2) holds in a weak sense since f ∈W1,nloc

(M); see [39, p. 136].This means that, for all n-forms η ∈ C

∞0

(M),

(3.7)

∫

M

〈f ∗d(V yω), η〉 =

∫

M

〈f ∗(V yω), δη〉,


where δ is the codifferential. Consequently, equations (3.4)–(3.6) are to be inter-preted in weak sense. In particular, combining (3.2), (3.3), and (3.7) we get

∫

M

〈JfWyω, δη〉 =

∫

M

〈f ∗(V yω), δη〉 =

∫

M

〈f ∗d(V yω), η〉 = 0

for all n-forms η ∈ C∞0

(M), and so (3.4) holds in weak sense.

Let us next specify the vector fieldW. Let A = x ∈M : Jf (x) = det f∗,x 6= 0.Hence f∗,x is invertible for all x ∈ A, and W = f

−1

∗ V in A. In M \A, either Jf (x)does not exist, which can happen only in a set of measure zero, or Jf (x) ≤ 0.Quasiregularity of f, more precisely the distortion condition (1.4), implies thatf∗,x = Txf = 0 for almost every such x. Hence f∗,x = 0 for a.e. x ∈ M \ A.Setting W = 0 in M \ A, we obtain

f∗(V yω) = 0 = Wy f

∗ω

a.e. in M \ A. Hence f ∗(V yω) = Wy f∗ω a.e. in M, and so (3.3) holds.

3.1. Sketch of the proof of Reshetnyak’s theorem. We shall use Theorem3.1 to sketch the proof of Reshetnyak’s theorem in a way that uses analysis, inparticular, A-harmonic functions. First we recall some definitions concerningp-capacity. If Ω ⊂ M is an open set and C ⊂ Ω is compact, then the p-capacityof the pair (Ω, C) is

(3.8) capp(Ω, C) = infϕ

∫

Ω

|∇ϕ|p,

where the infimum is taken over all functions ϕ ∈ C∞0

(Ω), with ϕ|C ≥ 1. Acompact set C ⊂M is of p-capacity zero, denoted by cappC = 0, if capp(Ω, C) =0 for all open sets Ω ⊃ C. Finally, a closed set F is of p-capacity zero, denotedby capp F = 0, if cappC = 0 for all compact sets C ⊂ F. It is a well-known factthat a closed set F ⊂ R

n containing a continuum C cannot be of n-capacity zero.This can be seen by taking an open ball B containing C and any test functionϕ ∈ C

∞0

(B), with ϕ|C = 1, and using a potential estimate

|ϕ(x) − ϕ(y)| ≤ c

(∫

B

|∇ϕ||x− z|n−1

dz +

∫

B

|∇ϕ||y − z|n−1

dz

), x, y ∈ B,

combined with a maximal function and covering arguments. Similarly, if C isa continuum in a domain Ω and B is an open ball, with B ⊂ Ω \ C, thencapn(C, B; Ω) > 0, where

capn(C, B; Ω) = infϕ

∫

Ω

|∇ϕ|p > 0,

the infimum being taken over all functions ϕ ∈ C∞(Ω), with ϕ|C = 1 and

ϕ|B = 0.


f is light. Suppose then that U ⊂ Rn is a domain and that f : U → R

n isa non-constant quasiregular mapping. We will show first that f is light whichmeans that, for all y ∈ R

n, the preimage f−1(y) is totally disconnected, i.e. each

component of f−1(y) is a point.

Fix y ∈ Rn and define u : R

n \ y → R by

u(x) = log1

|x− y| .

Then u is C∞ and n-harmonic in Rn \y by a direct computation. By Theorem

3.1, v = u f,v(x) = log

1

|f(x) − y| ,

is A-harmonic in an open non-empty set U \ f−1(y) and v(x) → +∞ as x→ z ∈f−1(y). We set v(z) = +∞ for z ∈ f

−1(y).

To show that f is light we use the logarithmic Caccioppoli inequality (2.4).Suppose that C ⊂ f

−1(y) ∩ U is a continuum. Since f is non-constant andcontinuous, there exists m > 1 such that the set Ω = x ∈ U : v(x) > m is anopen neighborhood of C and Ω ⊂ U. We choose another neighborhood D of Csuch that D ⊂ Ω is compact. Now we observe that vi = minv, i is a positivesupersolution for all i > m. The logarithmic Caccioppoli inequality (2.4) thenimplies that ∫

D

|∇ log vi|n ≤ c capn(Ω, D) ≤ c <∞

uniformly in i. Hence |∇ log v| ∈ Ln(D). Choose an open ball B such that

B ⊂ D \ f−1(y). We observed earlier that

capn(C, B;D) > 0

since C is a continuum. Let

MB = maxB

log v.

Now the idea is to use

min1,max0, 1

klog

v

MB

as a test function for capn(C, B;D) for every k ∈ N. We get a contradiction since

0 < capn(C, B;D) ≤ k−n‖∇ log v‖Lp(D) → 0

as k → ∞. Thus f−1(y) can not contain a continuum.

Differentiability a.e. Assume that f = (f1, . . . , fn) : U → Rn, U ⊂ R

n, is

quasiregular. Then coordinate functions fj are A-harmonic again by Theorem3.1, since functions x = (x1, . . . , xn) 7→ xj are n-harmonic. Now there are at leasttwo ways to prove that f is differentiable almost everywhere. For instance, sinceeach fj is A-harmonic, one can show by employing reverse Holder inequality

techniques that, in fact, f ∈ W1,ploc

(U), with some p > n. This then impliesthat f is differentiable a.e. in U ; see e.g. [3]. Another way is to conclude


that f is monotone, i.e. each coordinate function fj is monotone, and therefore

differentiable a.e. since f ∈ W1,nloc

(U); see [41]. The monotonicity of fj holdssince A-harmonic functions obey the maximum principle.

f is sense-preserving. Here one first shows that conditions f ∈ W1,nloc

(U) andJf (x) ≥ 0 a.e. imply that f is weakly sense-preserving, i.e. µ(y, f,D) ≥ 0 for alldomains D ⋐ U and for all y ∈ fD \ f∂D. This step employs approximation off by smooth mappings. Pick then a domain D ⋐ U and y ∈ fD \ f∂D. Denoteby Y the y-component of R

n \ f∂D and write V = D ∩ f−1Y . Since f is light,

D \ f−1(y) is non-empty. Thus we can find a point x0 ∈ f−1(y) ∩ V. Next we

conclude that the set x ∈ V : Jf (x) > 0 has positive measure. Otherwise, sincef is ACL and |f ′(x)| = 0 a.e. in V , f would be constant in a ball centered atx0 contradicting the fact that f is light. Thus there is a point x in V where f isdifferentiable and Jf (x) > 0. Now a homotopy argument, using the differentialof f at z, and µ(y, f,D) ≥ 0 imply that f is sensepreserving.

f is discrete and open. This part of the proof is purely topological. A sense-preserving light mapping is discrete and open by Titus and Young; see e.g. [41].

Further properties of f . Once Reshetnyak’s theorem is established it is pos-sible to prove further properties for quasiregular mappings. We collect theseproperties to the following theorems and refer to the books [39] and [41] for theproofs.

Theorem 3.2. Let f : M → N be a non-constant quasiregular map. Then

1. |fE| = 0 if and only if |E| = 0.2. |Bf | = 0, where Bf is the branch set of f, i.e. the set of all x ∈ M where

f does not define a local homeomorphism.

3. Jf (x) > 0 a.e.

4. The integral transformation formula∫

A

(h f)(x)Jf (x)dm(x) =

∫

N

h(y)N(y, f, A)dm(y)

holds for every measurable h : N → [0,+∞] and for every measurable A ⊂M, where N(y, f, A) = card f−1(y) ∩ A.

5. If u ∈ W1,nloc

(N,R), then v = u f ∈W1,nloc

(M,R) and

∇v(x) = TxfT∇u(f(x)) a.e.

Furthermore, we have a generalization of the morphism property.

Theorem 3.3. Let f : M → N be quasiregular and let u : N → R be an A-

harmonic function (or a subsolution or a supersolution, respectively) of type n.

Then v = uf is f#A-harmonic (a subsolution or a supersolution, respectively),

where

f#Ax(h) =

Jf (x)Txf

−1Af(x)(Txf−1)Th), if Jf (x) exists and is positive,

|h|n−2h, otherwise.


The ingredients of the proof of Theorem 3.3 include, for instance, the localityof A-harmonicity, Theorem 3.2, and a method to ”push-forward” (test) functions;see e.g. [16] and [41].

4. Modulus and capacity inequalities

Although the main emphasis of this survey is on the relation between quasireg-ular mappings and p-harmonic functions, we want to introduce also the othermain tool in the theory of quasiregular mappings. Let 1 ≤ p < ∞ and let Γbe a family of paths in M. We denote by F(Γ) the set of all Borel functions : M → [0,+∞] such that ∫

γ

ds ≥ 1

for all locally rectifiable path γ ∈ Γ. We call the functions in F(Γ) admissible

for Γ. The p-modulus of Γ is defined by

Mp(Γ) = inf∈F(Γ)

∫

M

pdm.

There is a close connection between p-modulus and p-capacity. Indeed, supposethat Ω ⊂ M is open and C ⊂ Ω is compact. Let Γ be the family of all paths inΩ \ C connecting C and ∂Ω. Then

(4.1) capp(Ω, C) = Mp(Γ).

The inequality capp(Ω, C) ≥ Mp(Γ) follows easily since = |∇ϕ| is admissiblefor Γ for each function ϕ as in (3.8). The other direction is harder and requiresan approximation argument; see [41, Proposition II.10.2].

If p = n = the dimension of M , we call Mn(Γ) the conformal modulus of Γ,or simply the modulus of Γ. In that case Mn(Γ) is invariant under conformalchanges of the metric. In fact, Mn(Γ) can be interpreted as follows: Define anew measurable Riemannian metric

〈〈·, ·〉〉 = 2〈·, ·〉.

Then, with respect to 〈〈·, ·〉〉, the length of γ has a lower bound

ℓ〈〈·,·〉〉(γ) =

∫

γ

ds ≥ 1

and the volume of M is given by

Vol〈〈·,·〉〉(M) =

∫

M

ndm.

Thus we are minimizing the volume of M under the constraint that paths in Γhave length at least 1.

The importance of the conformal modulus for quasiregular mappings lies inthe following invariance properties; see [41, II.2.4, II.8.1]


Theorem 4.1. Let f : M → N be a non-constant quasiregular mapping. Let

A ⊂ M be a Borel set with N(f,A) := supyN(y, f, A) < ∞, and let Γ be a

family of paths in A. Then

(4.2) Mn(Γ) ≤ KO(f)N(f,A)Mn(fΓ).

Theorem 4.2 (Poletsky’s inequality). Let f : M → N be a non-constant quasireg-

ular mapping and let Γ be a family of paths in M. Then

(4.3) Mn(fΓ) ≤ KI(f)Mn(Γ).

The proof of (4.2) is based on the change of variable formula for integrals(Theorem 3.2 3.) and on Fuglede’s theorem. The estimate (4.3) in the conversedirection is more useful than (4.2) but also much harder to prove; see [41, p.39–50].

As an application of the use of p-modulus and p-capacity, we prove a Harnack’sinequality for positive A-harmonic functions of type p > n − 1. Assume thatΩ ⊂ M is a domain, D ⋐ Ω another domain, and C ⊂ D is compact. Forp > n− 1, we set

λp(C,D) = infE,F

Mp(Γ(E,F ;D)),

where E and F are continua joining C and Ω \D, and Γ(E,F ;D) is the familyof all paths joining E and F in D.

Theorem 4.3 (Harnack’s inequality, p > n− 1). Let Ω, D, and C be as above.

Let u be a positive A-harmonic function in Ω of type p > n− 1. Then

(4.4) logMC

mC

≤ c0

(capp(Ω, D)

λp(C,D)

)1/p

,

where

MC = maxx∈C

u(x), mC = minx∈C

u(x),

and c0 = c0(p, β/α).

Ω

F D

E

Cγ

Proof. We may assume that MC > mC . Let ε > 0 be so small that MC − ε >

mC + ε. Then the sets x : u(x) ≥ MC − ε and x : u(x) ≤ mC + ε containcontinua E and F , respectively, that join C and Ω \D. Write

w =log u− log(mC + ε)

log(MC − ε) − log(mC + ε)


and observe that w ≥ 1 in E and w ≤ 0 in F. Therefore |∇w| is admissible forΓ(E,F ;D) and hence

∫

D

|∇w|p ≥Mp(Γ(E,F ;D)) ≥ λp(C,D).

On the other hand,∫

D

|∇ log u|pdm ≤ c(p, β/α) capp(Ω, D)

by the logarithmic Caccioppoli inequality (2.4), and

∇ log u =

(log

MC − ε

mC + ε

)∇w.

Hence

logMC − ε

mC + ε≤ c0

(capp(Ω, D)

λp(C,D)

)1/p

and (4.4) follows by letting ε→ 0.

We can define λp(C,D) analogously for p ≤ n − 1, too. However, λp(C,D)vanishes for p ≤ n − 1. Consequently, Theorem 4.3 is useful only for p > n− 1.The idea of the proof is basically due to Granlund [13]. In the above form, (4.4)appeared first time in [17]. In general, it is difficult to obtain an effective lowerbound for λp(C,D) together with an upper bound for capp(Ω, D). However, ifM = R

n and p = n, one obtains a global Harnack inequality by choosing C, D,and Ω as concentric balls.

5. Liouville-type results for A-harmonic functions

We say that M is strong p-Liouville if M does not support non-constantpositive A-harmonic functions for any A of type p. We have already mentionedthat a global Harnack inequality

maxB(x,r)

u ≤ c minB(x,r)

u

holds for non-negative A-harmonic functions onB(x, 2r) with a (Harnack-)constantc independent of x, r, and u ifM is complete and admits a global volume doublingcondition and (1, p)-Poincare’s inequality. It follows from the global Harnack in-equality that such manifold M is strong p-Liouville.

Example 5.1. 1. LetM be complete with non-negative Ricci curvature. Thenit is well-known that M admits a global volume doubling property by theBishop-Gromov comparison theorem (see [2], [8]). Furthermore, Buser’sisoperimetric inequality [6] implies that M also admits a (1, p)-Poincareinequality for every p ≥ 1. Hence M is strong p-Liouville.


2. Let Hn be the Heisenberg group. We write elements of Hn as (z, t), wherez = (z1, . . . , zn) ∈ C

n and t ∈ R. Furthermore, we assume that Hn isequipped with a left-invariant Riemannian metric in which the vector fields

Xj =∂

∂xj

+ 2yj∂

∂t,

Yj =∂

∂yj

− 2xj∂

∂t,

T =∂

∂t,

j = 1, . . . , n, form an orthonormal frame. Harnack’s inequality for non-negative A-harmonic functions on Hn was proved in [18] by using Jerison’sversion of Poincare’s inequality. Jerison proved in [31] that (1,1)-Poincare’sinequality holds for the horizontal gradient

∇0u =n∑

j=1

((Xju)Xj + (Yju)Yj)

and for balls in so-called Carnot-Caratheodory metric. Since the Lp-normof the Riemannian gradient is larger than that of the horizonal gradient,we have (1,1)-Poincare’s inequality for the Riemannian gradient as wellif geodesic balls are replaced by Carnot-Caratheodory balls or Heisenbergballs BH(r) = (z, t) ∈ Hn : (|z|4 + t

2)1/4< r and their left-translations.

Classically, a Riemannian manifold M is called parabolic if it does not supporta positive Green’s function for the Laplace equation.

Definition 5.2. We say that a Riemannian manifold M is p-parabolic, with1 < p <∞, if

capp(M,C) = 0

for all compact sets C ⊂M. Otherwise, we say that M is p-hyperbolic.

Example 5.3. 1. A compact Riemannian manifold is p-parabolic for all p ≥1.

2. In the Euclidean space Rn we have precise formulas for p-capacities of balls:

capp(Rn, B(r)) =

c r

n−p, if 1 ≤ p < n,

0, otherwise.

Hence Rn is p-parabolic if and only if p ≥ n.

3. If the Heisenberg group Hn is equipped with the left-invariant Riemannianmetric, we do not have precise formulas for capacities of balls. However,for r ≥ 1,

capp(Hn, BH(r)) ≈ r2n+2−p

if 1 ≤ p < 2n + 2, and capp(Hn, BH(r)) = 0 if p ≥ 2n + 2. Hence Hn isp-parabolic if and only if p ≥ 2n+ 2.


4. Any complete Riemannian manifold M with finite volume Vol(M) < ∞is p-parabolic for all p ≥ 1. This is easily seen by fixing a point o ∈ M

and taking a function ϕ ∈ C∞0

(B(o,R)

), with ϕ|B(o, r) = 1 and |∇ϕ| ≤

c/(R− r). We obtain an estimate

capp(B(o,R), B(o, r)

)≤ cVol(M)/(R− r)p → 0

as R → ∞.

5. Let Mn be a Cartan-Hadamard n-manifold, i.e. a complete, simply con-nected Riemannian manifold of non-positive sectional curvatures and di-mension n. If sectional curvatures have a negative upper bound KM ≤−a2

< 0, then M is p-hyperbolic for all p ≥ 1. This follows since Mn

satisfies an isoperimetric inequality

Vol(D) ≤ a

n− 1Area(∂D)

for all domains D ⋐ M, with smooth boundary; see [46], [7]. Anotherproof uses the Laplace comparison and Green’s formula. If p > 1, thenv(x) = exp(−δd(x, o)) is a positive supersolution of the p-Laplace equationfor some δ = δ(n, p) > 0 (see [20]). Hence the p-hyperbolicity of M alsofollows from the theorem below for p > 1.

Theorem 5.4. Let M be a Riemannian manifold and 1 < p < ∞. Then the

following conditions are equivalent:

1. M is p-parabolic.

2. Mp(Γ∞) = 0, where Γ∞ is the family of all paths γ : [0,∞) →M such that

γ(t) → ∞ as t→ ∞.

3. Every non-negative supersolution of

(5.1) − divAx(∇u) = 0

on M is constant for all A of type p.

4. M does not support a positive Green’s function g(·, y) for (5.1) for any Aof type p and y ∈M.

Here γ(t) → ∞ means that γ(t) eventually leaves any compact set. For theproof of Theorem 5.4 as well as for the discussion below we refer to [17].

Let us explain what is Green’s function for (5.1). We define it first in a”regular” domain Ω ⋐ M, where regular means that the Dirichlet problem forA-harmonic equation is solvable with continuous boundary data. For this notion,see [16]. We need a concept of A-capacity. Let C ⊂ Ω be compact, and assumefor simplicity that Ω \ C is regular. Thus there exists a unique A-harmonicfunction in Ω \C with continuous boundary values u = 0 on ∂Ω and u = 1 in C.Call u the A-potential of (Ω, C). We define

capA(Ω, C) =

∫

Ω

〈Ax(∇u),∇u〉.

ThencapA(Ω, C) ≈ capp(Ω, C)


and furthermore,

(5.2) capA(Ω1, C1) ≥ capA(Ω2, C2)

if C2 ⊂ C1 and/or Ω1 ⊂ Ω2. Note that this property is obvious for variationalcapacities but capA is not necessary a variational capacity.

The definition of Green’s function, and in particular its uniqueness when p =n, relies on the following observation.

Lemma 5.5. Let Ω ⋐ M be a domain and let C ⊂ Ω be compact such that Ω\Cis regular. Let u be the A-potential of (Ω, C). Then, for every 0 ≤ a < b ≤ 1,

capA(u > a, u ≥ b) =capA(Ω, C)

(b− a)p−1.

Definition 5.6. Suppose that Ω ⋐ M is a regular domain and let y ∈ Ω. Afunction g = g(·, y) is called a Green’s function for (5.1) in Ω if

1. g is positive and A-harmonic in Ω \ y,2. limx→z g(x) = 0 for all z ∈ ∂Ω,3.

limx→y

g(x) = capA(Ω, y)1/(1−p),

which we interpret to mean limx→y g(x) = ∞ if p ≤ n,

4. for all 0 ≤ a < b < capA(Ω, y)1/(1−p),

capA(g > a, g ≥ b) = (b− a)1/(1−p).

Theorem 5.7. Let Ω ⋐ M be a regular domain and y ∈ Ω. Then there exists a

Green’s function for (5.1) in Ω. Furthermore, it is unique at least if p ≥ n.

Monotonicity properties (5.2) of A-capacity and the so-called Loewner prop-erty, i.e. capnC > 0 if C is a continuum, are crucial in proving the uniquenesswhen p = n. Indeed, we can show that on sufficiently small spheres S(y, r)

|g(x, y) − capA(Ω, B(y, r))1/(1−n)| ≤ c, x ∈ S(y, r),

which then easily implies the uniqueness.

Next take an exhaustion of M by regular domains Ωi ⊂ Ωi+1 ⋐ M, M = ∪iΩi.

We can construct an increasing sequence of Green’s functions gi(·, y) on Ωi. Thenthe limit is either identically +∞ or

g(·, y) := limi→∞

gi(·, y)

is a positive A-harmonic function on M \ y. In the latter case we call the limitfunction g(·, y) a Green’s function for (5.1) on M.

We have the following list of Liouville-type properties of M (which may ormay not hold for M):

(1) M is p-parabolic.(2) Every non-negative A-harmonic function on M is constant for every A of

type p. (Strong p-Liouville.)


(3) Every bounded A-harmonic function on M is constant for every A of typep. (p-Liouville.)

(4) Every A-harmonic function u on M with ∇u ∈ Lp(M) is constant for every

A of type p. (Dp-Liouville.)

We refer to [17] for the proof of the following general result, and to [18] and[25] for studies concerning the converse directions.

Theorem 5.8.(1) ⇒ (2) ⇒ (3) ⇒ (4).

Next we discuss the close connection between the volume growth and p-parabolicity. Suppose that M is complete. Fix a point o ∈ M and writeV (t) = Vol

(B(o, t)

).

Theorem 5.9. Let 1 < p <∞ and suppose that∫ ∞ (

t

V (t)

)1/(p−1)

dt = ∞,

or ∫ ∞dt

V ′(t)1/(p−1)= ∞.

Then M is p-parabolic.

Proof. One can either construct a test function involving the integrals above, oruse a p-modulus estimate for separating (spherical) rings. More precisely, writeB(t) = B(o, t) and S(t) = S(o, t) = ∂B(o, t). For R > r > 0 and integers k ≥ 1,we write ti = r + i(R− r)/k, i = 0, 1, . . . , k. Then, by a well-known property ofmodulus,

Mp

(Γ(S(r), S(R); B(R)

))1/(1−p) ≥k−1∑

i=0

Mp

(Γ

(S(ti), S(ti+1); B(ti+1)

))1/(1−p);

see e.g. [41, II.1.5]. Here Γ(S(r), S(R); B(R)

)is the family of all paths joining

S(r) and S(R) in B(R). For each i = 0, . . . , k − 1 we have an estimate

Mp

(Γ

(S(ti), S(ti+1); B(ti+1)

))≤ (V (ti+1) − V (ti)) (ti+1 − ti)

−p.

Hence(5.3)

Mp

(Γ(S(r), S(R); B(R)

))1/(1−p) ≥k−1∑

i=0

(V (ti+1) − V (ti)

ti+1 − ti

)1/(1−p)

(ti+1 − ti).

Thus the right-hand side of (5.3) tends to the integral∫ R

r

dt

V ′(t)1/(p−1)

as k → ∞. We obtain an estimate

Mp

(Γ

(S(r), S(R); B(R)

))≤

(∫ R

r

dt

V ′(t)1/(p−1)

)1−p

.


In particular, if ∫ ∞

r

dt

V ′(t)1/(p−1)= ∞

for some r > 0, then M is p-parabolic.

The converse is not true in general. That is, M can be p-parabolic even if∫ ∞ (

t

V (t)

)1/(p−1)

dt <∞

or ∫ ∞dt

V ′(t)1/(p−1)<∞;

see [44].

It is interesting to study when the converse is true. We refer to [19] for theproofs of the following two theorems.

Theorem 5.10. Suppose that M is complete and admits a global doubling prop-

erty and global (1, p)-Poincare inequality for 1 < p <∞. Then

(5.4) M is p-hyperbolic if and only if

∫ ∞ (t

V (t)

)1/(p−1)

dt <∞.

In some cases, we can estimate Green’s functions:

Theorem 5.11. Suppose that M is complete and has non-negative Ricci curva-

ture everywhere. Let 1 < p <∞. Then

M is p-hyperbolic if and only if

∫ ∞ (t

V (t)

)1/(p−1)

dt <∞.

Furthermore, we have estimates for Green’s functions for (5.1)

c−1

∫ ∞

2r

(t

V (t)

)1/(p−1)

dt ≤ g(x, o) ≤ c

∫ ∞

2r

(t

V (t)

)1/(p−1)

dt

for every x ∈ ∂M(r), where M(r) is the union of all unbounded components of

M \ B(o, r). The constant c depends only on n, p, α, and β.

Theorem 5.10 follows also from the following sharper result; see [21].

Theorem 5.12. Suppose that M is complete and that there exists a geodesic ray

γ : [0,∞) →M such that for all t > 0,

|B(γ(t), 2s)| ≤ c|B(γ(t), s)|,whenever 0 < s ≤ t/4, and that

∫

Bγ(t)

|u− uBγ(t)|dm ≤ c

∫

2Bγ(t)

|∇u|pdm

1/p


for all u ∈ C∞(

2Bγ(t)), where Bγ(t) = B(γ(t), t/8). Then M is p-hyperbolic if

∫ ∞ (t

|B(γ(t), t/4)|

)1/(p−1)

dt <∞.

Theorem 5.12 can be applied to obtain the following.

Theorem 5.13. Let M be a complete Riemannian n-manifold whose Ricci curva-

ture is non-negative outside a compact set. Suppose that M has maximal volume

growth (V (t) ≈ rn). Then M is p-parabolic if and only if p ≥ n.

To our knowledge it is an open problem whether the equivalence (5.4) holdsfor a complete Riemannian n-manifold whose Ricci curvature is non-negativeoutside a compact set.

6. Liouville-type results for quasiregular mappings

Here we give applications of the above results on n-parabolicity and variousLiouville properties to the existence of non-constant quasiregular mappings be-tween given Riemannian manifolds.

Let us start with the Gromov-Zorich ”global homeomorphism theorem” thatis a generalization of Zorich’s theorem we mentioned in the introduction; see [14],[48].

Theorem 6.1. Suppose that M is n-parabolic, n = dimM ≥ 3, and that N is

simply connected. Let f : M → N be a locally homeomorphic quasiregular map.

Then f is injective and fM is n-parabolic.

Proof. We give here a very rough idea of the proof. First one observes thatfM is n-parabolic (see Theorem 6.2 below), and so N \ fM is of n-capacityzero. Then one shows, again by using the n-parabolicity of M, that the set Eof all asymptotic limits of f is of zero capacity. Consequently, E is of Hausdorffdimension zero. Recall that an asymptotic limit of f is a point y ∈ N such thatf

(γ(t)

)→ y as t → ∞ for some path γ ∈ Γ∞ in M. Removing E ∪ (N \ FM)

from N has no effect on the simply connectivity for dimensions n ≥ 3. That is,fM \E remains simply connected. Thus one can extend uniquely any branch oflocal inverses of f and obtain a homeomorphism g : fM \ E → g(fM \ E) suchthat f g = id |(fM \E). Finally, g can be extended to E to obtain the inverseof f.

In [22] we generalized the global homeomorphism theorem for mappings offinite distortion under mild conditions on the distortion. See also [49] for arelated result for locally quasiconformal mappings.

Theorem 6.2. If N is n-hyperbolic and M is n-parabolic, then every quasiregular

mapping f : M → N is constant.


Proof. Suppose that f : M → N is a non-constant quasiregular mapping. ThenfM ⊂ N is open. If fM 6= N, pick a point y ∈ ∂(N \ fM) and let g =g(·, y) be the Green’s function on N for the n-Laplacian. Then g f is a non-constant positive A-harmonic function on M which gives a contradiction withthe n-parabolicity of M and Theorem 5.8. If fM = N, let u be a non-constantpositive supersolution on N for the n-Laplacian. Then u f is a non-constantsupersolution on M for some A of type n which is again a contradiction.

Example 6.3. 1. If N is a Cartan-Hadamard manifold, with KN ≤ −a2< 0,

then every quasiregular mapping f : Rn → N is constant.

2. Let Hn be the Heisenberg group with a left-invariant Riemannian metric,then every quasiregular mapping f : R

2n+1 → Hn is constant.

Theorem 6.4. Suppose that M is strong n-Liouville while N is not. Then every

quasiregular map f : M → N is constant.

Proof. If N is not strong n-Liouville, then it is n-hyperbolic by Theorem 5.8.Suppose that f : M → N is a non-constant quasiregular mapping. Then fM ⊂N is open. If fM 6= N, choose a point y ∈ ∂(N \ fM) and let g = g(·, y)be the Green’s function for n-Laplacian on N . Then g f is a non-constantpositive A-harmonic function, with A of type n. This is a contradiction. IffM = N , we choose a non-constant positive n-harmonic function u on N andget a contradiction as above.

Theorem 6.5. Let N be a Cartan-Hadamard n-manifold, with −b2 ≤ K ≤−a2

< 0, and let M be a complete Riemannian n-manifold admitting a global

doubling property and a global (1, n)-Poincare inequality. Then every quasiregular

mapping f : M → N is constant.

Proof. By [20], N admits non-constant positive n-harmonic functions. HenceN is not strong n-Liouville. On the other hand, the assumptions on M implythat a global Harnack’s inequality for positive A-harmonic functions of type nholds on M. Thus M is strong n-Liouville, and the claim follows from Theorem6.4.

Theorem 6.6 (”One-point Picard”). Suppose that N is n-hyperbolic and M is

strong n-Liouville. Then every quasiregular mapping f : M → N \ y, with

y ∈ N, is constant.

Proof. Suppose that f : M → N \ y is a non-constant quasiregular mapping.Then ∂(N \ fM) 6= ∅. Choose a point z ∈ ∂(N \ fM), and let g = g(·, z) be theGreen’s function on N for the n-Laplacian. Then g f is a non-constant positiveA-harmonic function for some A of type n leading to a contradiction.

7. Picard-type theorems

The classical big Picard theorem states that a holomorphic mapping of thepunctured unit disc z ∈ C : 0 < |z| < 1 into the complex plane omitting two


values has a meromorphic extension to the whole disc; see e.g. [1, Theorem 1-14]. In [40] Rickman proved a counterpart of Picard’s theorem for quasiregularmappings (Theorem 7.1) and its local version (Theorem 7.2) corresponding tothe big Picard theorem.

Theorem 7.1 ([40]). For each integer n ≥ 2 and each K ≥ 1 there exists

a positive integer q = q(n,K) such that if f : Rn → R

n \ a1, . . . , aq is K-

quasiregular and a1, . . . , aq are distinct points in Rn, then f is constant.

Theorem 7.2 ([40]). Let G = x ∈ Rn : |x| > s and let f : G → R

n \a1, . . . , aq be a K-quasiregular mapping, where a1, . . . , aq are distinct points in

Rn and q = q(n,K) is the integer in Theorem 7.1. Then the limit lim|x|→∞ f(x)

exists.

In this section we consider corollaries and extensions of the big Picard theoremfor quasiregular mappings.

Although the short argument yielding Theorem 7.1 from Theorem 7.2 is well-known, it seems that the following corollary employing the same argument hasgone unnoticed in the literature.

Corollary 7.3. Let K ≥ 1 and R > 0. Let f : Rn → R

n be a continuous mapping

omitting at least q = q(n,K) points, where q(n,K) is as in Theorem 7.1. Then

at least one of the following conditions fails:

(i) f |Rn \ Bn(R) is K-quasiregular,

(ii) fBn(r) is open for some r > R.

Proof. Suppose towards a contradiction that both conditions (i) and (ii) hold.By Theorem 7.2, the mapping f has a limit at the infinity. Hence we mayextend f to a continuous mapping R

n → Rn. Moreover, f is K-quasiregular

in Rn \ Bn(R). By composing f with a Mobius mapping if necessary, we may

assume that f(∞) = ∞. Since f is a non-constant quasiregular mapping onRn \ Bn(R), f |Rn \ Bn(R) is an open mapping. Hence fR

n is open in Rn, by (ii).

Since fRn is both open and closed, fR

n = Rn and fR

n = Rn. This contradicts

the assumption that f omits q points. The claim follows.

In [23] the authors consider quasiregular mappings of the punctured unit ballinto a Riemannian manifold N . We say that N has at least q ends, if there existsa compact set C ⊂ N such that N \ C has at least q components which are notrelatively compact. Such a component of N \C is called an end of M with respect

to C. Let E be the set of ends of N , that is, E ∈ E is an end of N with respect tosome compact set C ⊂ N . We compactify N with respect to its ends as follows.There is a natural partial order in E induced by inclusion. We call a maximaltotally ordered subset of E an asymptotic end of N . The set of asymptotic endsof N is denoted by ∂N and N = N ∪ ∂N . We endow N with a topology suchthat the inclusion N ⊂ N is an embedding and for every e ∈ ∂N sets E ∈ e

form a neighborhood basis at e. The main result is the following version of thebig Picard theorem.


Theorem 7.4 ([23, Theorem 1.3]). For every K ≥ 1 there exists q = q(K,n)such that every K-quasiregular mapping f : Bn\0 → N has a limit limx→0 f(x)

in N if N has at least q ends.

In the spirit of Corollary 7.3 we formulate the following consequence Theorem7.4.

Corollary 7.5. Given n ≥ 2 and K ≥ 1 there exists q = q(n,K) such that the

following holds. Suppose that M is compact, z1, . . . , zk ⊂M , where 1 ≤ k < q,

and that N has at least q ends. Let f : M \ z1, . . . , zk → N be a continuous

mapping and let Ωi be a neighborhood of zi for every 1 ≤ i ≤ k. Then at least

one of the following conditions fails:

(i) f is K-quasiregular in Ωi \ zi for every i,

(ii) there exists a neighborhood Ω of M \ (Ω1 ∪ · · · ∪ Ωk) such that fΩ is open.

Proof. Suppose that both conditions are satisfied. For every 1 ≤ i ≤ k wefix a 2-bilipschitz chart ϕi : Ui → ϕiUi at zi. We may assume that Ui ⊂ Ωi.Every mapping f ϕ−1

i |ϕi(Ui \ zi) is 2nK-quasiregular, and therefore it has alimit at ϕi(zi) by Theorem 7.4 if N has at least q(n, 2nK) ends. Hence f has

a limit at every point zi. We extend f to a continuous mapping f : M → N .Denote M ′ = M \ z1, . . . , zk. Since f is an open mapping, ∂fM ′ ∩ fM ′ = ∅.Furthermore, since M is compact,

fM ′ ⊂ fM = fM.

Hence ∂fM ′ ⊂ fM \ fM ′. Thus card(∂fM ′) ≤ card(fM \ fM ′) ≤ k and

N = N = fM ′ = fM.

This is a contradiction, since

card(fM \ fM ′) ≤ k < q ≤ card(N \ fM ′) = card(fM \ fM ′).

In [26] Holopainen and Rickman applied a method of Lewis ([33]) that relies onHarnack’s inequality to prove the following general version of Picard’s theorem onthe number of omitted values of a quasiregular mapping. We say that a completeRiemannian n-manifold M belongs to the class M(m,ϑ), where m : (0, 1) → N

and ϑ : (0,∞) → (0,∞) are given functions, if following two conditions hold:

(m) for each 0 < λ < 1 every ball of radius r in M can contain at most m(λ)disjoint balls of radius λr, and

(ϑ) M admits a global Harnack’s inequality for non-negative A-harmonic func-tions of type n with Harnack-constant ϑ(β/α), where α and β are theconstants of A.

Theorem 7.6 ([26]). Given n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0,∞) →(0,∞) there exists q = q(n,K,m, ϑ) ≥ 2 such that the following holds. Suppose

that M belongs to the class M(m,ϑ) and that N has at least q ends. Then every

K-quasiregular mapping f : M → N is constant.


Next we show that this theorem admits a local version. Suppose that M iscomplete. We say that an asymptotic end e of M is of type E(m,ϑ) if thereexists E ∈ e such that

(Em) for each 0 < λ < 1 every ball of radius r in E can contain at most m(λ)disjoint balls of radius λr, and

(Eϑ) E admits a uniform Harnack inequality for non-negative A-harmonic func-tions of E of type n for balls B ⊂ E satisfying 4B ⊂ E. We also assumethat the Harnack constant ϑ depends only on β/α, where α and β are theconstants of A.

We also say that an asymptotic end e of M is p-parabolic (with p ≥ 1) if thereexists E ∈ e such that for every ε > 0 there exists E ′ ∈ e such that

Mp(Γ(E ′,M \ E;M)) < ε.

Furthermore, we say that an asymptotic end e of M is locally C-quasiconvex

if for every E ∈ e there exists E ′ ∈ e, E′ ⊂ E, such that each pair of points

x, y ∈ E′ can be joint by a path in E ′ of length at most Cd(x, y), where d is the

Riemannian distance of M .

Theorem 7.7. Let n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0,∞) → (0,∞).Then there exists q = q(n,K,m, ϑ) such that the following holds. Suppose that

M is complete and e is an n-parabolic locally C-quasiconvex asymptotic end of

M of type E(m,ϑ), and that N has at least q ends. Let E ∈ e and f : E → N be

a K-quasiregular mapping. Then f has a limit at e.

Corollary 7.8. Let n ≥ 2, K ≥ 1, m : (0, 1) → N, and ϑ : (0,∞) → (0,∞).Then there exists q = q(n,K,m, ϑ) such that the following holds. Suppose that

a complete Riemannian n-manifold M has asymptotic ends e1, . . . , ek, k < q,

of type E(m,ϑ) which are all n-parabolic and locally C-quasiconvex, and that N

has at least q ends. Let f : M → N be a continuous mapping and Ei ∈ ei for

every 1 ≤ i ≤ k. Then at least one of the following conditions fails:

(i) f is K-quasiregular in Ei for every i,

(ii) there exists a neighborhood Ω of M \ (E1 ∪ · · · ∪Ek) such that fΩ is open.

Proof. Suppose that both conditions hold. By Theorem 7.7, we may extend f

to a continuous mapping f : M → N . Since M is compact, we may follow theproof of Corollary 7.5.

We need several lemmas in order to prove Theorem 7.7. Let us first recallthe definition of a Harnack function. Let M be a Riemannian manifold. Acontinuous function u : M → R is called a Harnack function with constant θ if

M(h, x, r) := supB(x,r)

h ≤ θ infB(x,r)

h

holds in each ball B(x, r) whenever the function h is nonnegative in B(x, 2r),has the form h = ±u + a for some a ∈ R, and B(x, 2r) ⊂ M is compact. Theoriginal version of Lewis’ lemma is stated for Harnack functions. It is well known


(see [16, 6.2]) that A-harmonic functions in the Euclidean setting are Harnackfunctions with some θ depending only on n and on the constants p, α, and β ofA. In that case θ is called the Harnack constant of A.

Lemma 7.9. Let e be an n-parabolic locally C-quasiconvex asymptotic end of a

complete Riemannian n-manifold M . Suppose u : E → R, where E ∈ e, is a Har-

nack function with constant θ such that lim supx→e u(x) = ∞ and lim infx→e u(x) <0. Then for every C0 > 0 there exists a ball B = B(x0, r0) ⊂ E such that

(1) B(x0, 100Cr0) ⊂ E,

(2) u(x0) = 0, and

(3) maxB u ≥ C0.

Proof. It is sufficient to modify the proof of [23, Lemma 2.1] as follows. LetE

′ ∈ e be such that E ′ ⊂ E and E ′ is C-quasiconvex. Let F ′ ⊂M be a compactset such that E ′ is a component of M \ F ′, fix o ∈ M , and let R0 > 0 be suchthat F ′ ⊂ B(o,R0/2).

Fix k ∈ N such that given r > R0 and x, y ∈ ∂B(o, r)∩E ′ there exists k ballsBi = B(xi, r/1000), 1 ≤ i ≤ k, in E such that

(1) x ∈ B1,(2) y ∈ Bk,(3) xi ∈ E

′ for every i, and(4) Bi ∩Bi+1 6= ∅ for every i ∈ 1, . . . , k − 1.

Indeed, since Bi∩B(o,R0/2) = ∅, we have Bi ⊂ E, and since E ′ is C-quasiconvex,we may choose any k > 2000C. We may now apply the proof of [23, Lemma 5]almost verbatim.

Lemma 7.10. Let e be an n-parabolic locally C-quasiconvex asymptotic end of a

complete Riemannian n-manifold M and E ∈ e. If f : E → N is a quasiregular

mapping such that fE is n-hyperbolic, then f has a limit in N at e.

Proof. Suppose that fE is n-hyperbolic. We may assume thatE is C-quasiconvex.If f has no limit at e, there exists a compact set F ⊂ N such that fE ′ ∩ F 6= ∅for every E

′ ∈ e. Hence there exists a sequence (xk) such that xk → e andf(xk) → z ∈ N as k → ∞. Let (yk) be another sequence such that yk → e ask → ∞. We show that the hyperbolicity of fE yields f(yk) → z as k → ∞,which is a contradiction.

For every k we fix a path αk : [0, 1] → E such that αk(0) = xk, αk(1) = yk,and ℓ(αk) ≤ Cd(xk, yk). Then

capn(E, |αk|) → 0

as k → ∞. By Poletsky’s inequality (4.3) and (4.1),

capn(fE, f |αk|) ≤ KI(f) capn(E, |αk|)for every k. Suppose that f(yk) 6→ z. Then, by passing to a subsequenceif necessary, we may assume that d(f(yk), z) ≥ δ > 0 for every k. Since


d(f(αk(0)), f(αk(1))) ≥ δ/2 for large k, we have, by the n-hyperbolicity of fE,that

capn(fE, f |αk|) ≥ ε > 0.

for every k. This is a contradiction.

The following lemma is a reformulation of [29, Lemma 19.3.2].

Lemma 7.11. Let E ⊂ M , let u : E → R be a non-constant Harnack function

with constant θ, and let α : [a, b] → E be a path. If ℓ(α) ≤ k dist(|α|, u−1(0) ∪M \ E), then u has a constant sign on |α|. Furthermore,

max|α|

u ≤ θk min

|α|u

if u is positive on |α|, and

max|α|

u ≤ θ−k min

|α|u

if u is negative on |α|.

Proof. Since |α| is connected, every non-vanishing function on |α| has constantsign. We may assume without loss of generality that u is positive on |α|. Leta = a0 < a1 < . . . < ak = b be a partition of [a, b] such that ℓ(α|[ai, ai+1]) =ℓ(α)/k for every i = 0, 1, . . . , k − 1. For every i fix xi ∈ α([ai, ai+1]) such thatℓ(α|[ai, xi]) = ℓ(α|[xi, ai+1]). Then α([ai, ai+1]) ⊂ B(xi, ℓ(α)/(2k)). Further-more, B(xi, ℓ(α)/k) ⊂ E and B(xi, ℓ(α)/k) ∩ u

−1(0) = ∅. Since α(ai+1) ∈B(xi, ℓ(α)/(2k)) ∩ B(xi+1, ℓ(α)/(2k)) for every i = 1, . . . , k − 1, a repeated useof Harnack’s inequality yields max|α| u ≤ θ

k min|α| u.

Lemma 7.12 (Lewis’ lemma). Let M , e, E, and u be as in Theorem 7.7. Then

for every C0 > 0 there exists a ball B = B(x0, r0) ⊂ E such that

(1) 6B ⊂ E,

(2) u(x0) = 0, and

(3) C0 ≤ max6B u ≤ θ6 maxB u.

Proof. Let C0 > 0 and B(x0, R) be as in Lemma 7.9. Let Z = u−1(0) and

ZR = Z ∩ B(x0, 41R). For each x ∈ ZR we set rx = R − d(x, x0)/41 andBx = B(x, rx). Then F =

⋃x∈ZR

Bx is compact and x 7→ maxBxu is continuous.

Let a ∈ ZR be a point of maximum for this function. Thus

maxB(a,ra)

u ≥ maxB(x0,R)

u ≥ C0.

As in [29, Lemma 19.4.1], we have that

dist(Z, B(a, 6ra) \ F ) ≥ 5ra6.

Let y0 ∈ B(a, 6ra) be such that

u(y0) = maxB(a,6ra)

u ≥ C0 > 0.


If y0 ∈ F , then, by the maximal property of ball B(a, ra),

maxB(a,6ra)

u = u(y0) ≤ maxF

u = maxB(a,ra)

u ≤ θ6 maxB(a,ra)

u.

If y0 6∈ F , let y1 ∈ F∩B(a, 6ra) be nearest to y0 in length metric. As B(a, ra) ⊂ F

it follows that

d(y0, y1) ≤ dist(y0, B(a, ra)) ≤ 6ra − ra = 5ra.

Let α : [0, 1] → E be a path of minimal length such that α(0) = y0 and α(1) = y1.Then α[0, 1) ∩ F = ∅. Hence

dist(Z, |α|) ≥ 5ra6.

Thus ℓ(α) ≤ Cd(y0, y1) ≤ 6C dist(Z, |α|). By Lemma 7.11,

u(y0) ≤ max|α|

u ≤ θ6C min

|α|u ≤ θ

6Cu(y1) ≤ θ

6C maxF

u = θ6C max

B(a,ra)

u.

Lemma 7.13 ([24],[26]). Let N be an n-parabolic Riemannian manifold. Suppose

that C ⊂ N is compact such that N has q ends V1, . . . , Vq with respect to C. Then

there exist n-harmonic functions vj, j = 2, . . . , q, and a positive constant κ such

that

|vj| ≤ κ in C,(7.1)

|vj − vi| ≤ 2κ in V1,(7.2)

supV1

vj = ∞,(7.3)

infVj

vj = −∞,(7.4)

vj is bounded in Vk for k 6= 1, j,(7.5)

if vj(x) > κ, then x ∈ V1,(7.6)

if vj(x) < −κ then x ∈ Vj.(7.7)

Proof of Theorem 7.7. Suppose that a K-quasiregular mapping f : E → N

has no limit at e. By Lemma 7.10, N is n-parabolic. Let C ⊂ N be a compactset such that N has q ends V1, . . . , Vq with respect to C. For every j = 2, . . . , qlet us fix an n-harmonic function vj with properties (7.1) - (7.7) given in Lemma7.13. For every j = 2, . . . , q we set uj = vj f . Then functions uj are A-harmonicin E. Next we show that

(7.8) lim supx→e

uj(x) = +∞ and lim infx→e

uj(x) = −∞,

and hence they satisfy the assumptions of Lemma 7.9. This can be seen byobserving that the sets x ∈ N : vj(x) > c and x ∈ N : vj(x) < −c are non-empty and open for every c > 0 and j = 2, . . . , q. By Lemma 7.10, f(E \ F )intersects these sets for every compact F ⊂ M , and therefore (7.8) follows. By


Lemma 7.12 there are sequences xi ∈ E and ri ∈ (0,∞), i ∈ N, such thatu2(xi) = 0, B(xi, 3ri) ⊂ E,

M(u2, xi, 3ri) ≤ θ6M(u2, xi, ri/2),

andM(u2, xi, ri/2) → ∞ as i→ ∞. Let us fix an index i such thatM(u2, xi, ri/2) ≥4θκ, where θ > 1 is the Harnack constant of A and κ is the constant in Lemma7.13. We write x = xi and r = ri. By (7.6), f

(B(x, r/2)

)∩ V1 6= ∅. Thus, by

(7.2), we have

(7.9) M(u2, x, s) − 2κ ≤M(uj, x, s) ≤M(u2, x, s) + 2κ

whenever s ≥ r/2. Next we conclude by using Harnack’s inequality that

(7.10) M(uj, x, r) ≤ (θ − 1)M(−uj, x, 2r)for all j. Let us first show that uj(z) = 0 for some z ∈ B(x, r). Suppose onthe contrary, that uj > 0 in B(x, r). Then uj(y) ≤ θuj(x) for all y ∈ B(x, r/2)by Harnack’s inequality. Since M(u2, x, r/2) ≥ 4θκ, there exists y ∈ B(x, r/2)such that uj(y) > 2θκ by (7.9). Thus uj(x) > 2κ, and so x ∈ V1. By (7.2),u2(x) ≥ uj(x)− 2κ > 0 contradicting the assumption u2(x) = 0. Therefore thereexists z ∈ B(x, r) such that uj(z) = 0. Thus infB(x,r) uj ≤ 0. Inequality (7.10)follows now from the calculation

M(uj, x, r) = supB(x,r)

uj = supB(x,r)

(uj − inf

B(x,2r)uj

)+ inf

B(x,2r)uj

≤ θ infB(x,r)

(uj − inf

B(x,2r)uj

)+ inf

B(x,2r)uj

= θ infB(x,r)

uj + (1 − θ) infB(x,2r)

uj

≤ −(θ − 1) infB(x,2r)

uj = (θ − 1) supB(x,2r)

(−uj)

= (θ − 1)M(−uj, x, 2r),since uj − infB(x,2r) uj ≥ 0 in B(x, 2r).

Inequalities (7.9) and (7.10), and the assumptionM(u2, x, r/2) ≥ 4θκ togetheryield the inequality

(7.11) M(u2, x, r) ≤ θM(−uj, x, 2r).Indeed,

M(u2, x, r) ≤ M(uj, x, r) + θ−1M(u2, x, r)

≤ (θ − 1)M(−uj, x, 2r) + θ−1M(u2, x, r),

which is equivalent to (7.11). We fix zj ∈ B(x, 2r) such that

(7.12) M(−uj, x, 2r) = −uj(zj).The well-known oscillation estimate (see e.g. [16, 6.6])

oscB(y,ρ)

uj ≤ c(ρ/r)γ oscB(y,r)

uj


together with [24, Lemma 4.2] and (7.9) imply that

(7.13) oscB(zj ,ρ)

uj ≤ c1(ρ/r)γM(u2, x, 3r)

for ρ ∈ (0, r). See [24, (5.5)] for details. Thus

maxB(zj ,ρ)

uj = oscB(zj ,ρ)

uj + minB(zj ,ρ)

uj

≤ c1(ρ/r)γM(u2, x, 3r) + uj(zj)

≤ c1(ρ/r)γM(u2, x, 3r) − θ

−1M(u2, x, r),

by (7.13), (7.12), and (7.11). Since M(u2, x, 3r) ≤ θ6M(u2, x, r), we obtain

c1(ρ/r)γM(u2, x, 3r) ≤ (2θ)−1

M(u2, x, r)

by choosing ρ = (2θ7c1)

−1/γr. Hence

maxB(zj ,ρ)

uj ≤ −(2θ)−1M(u2, x, r) ≤ −2κ.

By (7.7), we conclude that f(B(zj, ρ)

)⊂ Vj and hence the balls B(zj, ρ) are

disjoint. Since B(zj, ρ) ⊂ B(x, 3r), there can be at most m(ρ/3r) of them.Hence q has an upper bound that depends only on n, K, ϑ, and m.

8. Quasiregular mappings, p-harmonic forms, and deRham cohomology

The use of n-harmonic functions in studying Liouville-type theorems for quasireg-ular mappings f : M → N is restricted to the case, where N is non-compact. Thereason for this restriction is simple: a compact Riemannian manifold does notcarry non-constant p-harmonic functions. Therefore, in the case of a compacttarget manifold, we have to use p-harmonic forms. In this final section we discussbriefly p-harmonic and A-harmonic forms and their connections to quasiregularmappings. For detailed discussions on A-harmonic forms, see e.g. [27], [28], [29],[30], and [42]. For the connection of A-harmonic forms to quasiregular mappings,see e.g. [4], [29], and [34].

The Riemannian metric of M induces an inner product to the exterior bundle∧ℓT

∗M for every ℓ ∈ 1, . . . , n, see e.g. [29, 9.6] for details. We denote this

inner product by 〈·, ·〉 and the corresponding norm by | · |. As usual, sections of

the bundle∧ℓ

T∗M are called ℓ-forms. The Lp-space of measurable ℓ-forms is

denoted by Lp(∧ℓ

M) and the Lp-norm is defined by

‖ξ‖p =

(∫

M

|ξ|pdx)1/p

.

The local Lp-spaces of ℓ-forms are denoted by Lploc

(∧ℓ

M). The space of C∞-

smooth ℓ-forms on M is denoted by C∞(

∧ℓM), and the space of compactly

supported C∞-smooth ℓ-forms by C∞0

(∧ℓ

M).


Let ℓ ∈ 1, . . . , n−1 and p > 1. Let A :∧ℓ

T∗M → ∧ℓ

T∗M be a measurable

bundle map such that there exists positive constants a and b satisfying

〈A(ξ) −A(ζ), ξ − ζ〉 ≥ a(|ξ| + |ζ|)p−2|ξ − ζ|2,(8.1)

|A(ξ) −A(ζ)| ≤ b(|ξ| + |ζ|)p−2|ξ − ζ|, and(8.2)

A(tξ) = t|t|p−2A(ξ)(8.3)

for all ξ, ζ ∈ ∧ℓT

∗xM , t ∈ R, and for almost every x ∈ M . We also assume that

x 7→ Ax(ω) is a measurable ℓ-form for every measurable ℓ-form ω : M → ∧ℓT

∗M .

We say that an ℓ-form ξ is A-harmonic (of type p) on M if ξ is a weakly closed

continuous form in W d,ploc

(∧ℓ

M) and satisfies equality

δ(A(ξ)) = 0

weakly, that is, ∫

M

〈A(ξ), dϕ〉 = 0

for all ϕ ∈ C∞0

(∧ℓ−1

M). Here Wd,ploc

(∧ℓ

M) is the partial Sobolev space of

ℓ-forms. A form ω ∈ Lploc

(∧ℓ

M) is in the space W d,ploc

(∧ℓ

M) if the distribu-

tional exterior derivative dω exists and dω ∈ Lploc

(∧ℓ+1

M). The global space

Wd,p(

∧ℓM) is defined similarly. A form ω ∈ W

d,ploc

(∧ℓ

M) is weakly closed if

dω = 0 and weakly exact if ω = dτ for some τ ∈ Wd,ploc

(∧ℓ−1

M).

Apart from minor differences between conditions (8.1)-(8.3) and the corre-sponding conditions in Section 2, we can say that A-harmonic functions corre-spond to A-harmonic weakly exact 1-forms.

Let f : M → N be a quasiregular mapping. Since f is almost everywhere

differentiable, we may define the pull-back f ∗ξ of the form ξ ∈ L

n/ℓ

loc(∧ℓ

N) by

(f ∗ξ)x = (Txf)∗ξf(x).

By the quasiregularity of f , f ∗ξ ∈ L

n/ℓ

loc(∧ℓ

M). Furthermore, d(f ∗ξ) = f

∗(dξ)

if ξ ∈ W1,n/ℓ

loc(∧ℓ

N). Hence f ∗ξ ∈ W

1,n/ℓ

loc(∧ℓ

M) for ξ ∈ W1,n/ℓ

loc(∧ℓ

M). Thequasiregularity of f also yields that the pull-back f ∗

ξ of an (n/ℓ)-harmonic ℓ-form

is A-harmonic. Similarly to the case of A-harmonic functions, A :∧ℓ

T∗M →∧ℓ

T∗M is defined by

A(η) = 〈G∗η, η〉(n/ℓ)−2

G∗η,

whereGx = Jf (x)

2/n(Txf)−1((Txf)−1

)Ta.e. .

Recently in [4] Bonk and Heinonen studied cohomology of quasiregularly el-liptic manifolds using p-harmonic forms. A connected Riemannian manifold iscalled K-quasiregularly elliptic if it receives a non-constant K-quasiregular map-ping from R

n. The main result of [4] is the following theorem.

Theorem 8.1 ([4, Theorem 1.1]). Given n ≥ 2 and K ≥ 1 there exists a constant

C = C(n,K) > 1 such that dimH∗(N) ≤ C for every K-quasiregularly elliptic

closed n-manifold N .


As the Picard-type theorem 7.6, also this theorem has a local counterpart.

Theorem 8.2 ([34, Theorem 2]). Given n ≥ 2 and K ≥ 1 there exists a constant

C′ = C

′(n,K) > 1 such that every K-quasiregular mapping f : Bn\0 → N has

a limit at origin if N is closed, connected, and oriented Riemannian n-manifold

with dimH∗(N) ≥ C

′.

We close this section with a sketch of the proof of Theorem 8.2. The followingtheorem on exact A-harmonic forms is essential in the proof. For details, see[34].

Theorem 8.3. Let n ≥ 3 and let η be a weakly exact A-harmonic ℓ-form, ℓ ∈2, . . . , n− 1, on R

n \ Bn such that

(8.4)

∫

Rn\Bn(2)

|η|n/ℓ = ∞.

Then there exists γ = γ(n, a, b) > 0 such that

(8.5) lim infr→∞

1

rγ

∫

Bn(r)\Bn(2)

|η|n/ℓ > 0.

Here a and b are as in (8.1) and (8.2).

Sketch of the proof of Theorem 8.2. Let us first consider some exceptions.For Riemannian surfaces the result is classical and follows from the uniformiza-tion theorem and the measurable Riemann mapping theorem, see [34, Theorem3]. For n ≥ 3 we may give a bound for the first cohomology using a well-knownresult of Varopoulos on the fundamental group and n-hyperbolicity. For details,see [34, Theorem 4]. Hence we may restrict our discussion to dimensions n ≥ 3and to cohomology dimensions ℓ ≥ 2.

Let n ≥ 3 and 2 ≤ ℓ ≤ n − 1, and suppose that f : Bn \ 0 → N does nothave a limit at the origin. Without changing the notation we precompose f witha sense-preserving Mobius mapping σ such that σ(Rn \ Bn) = B

n \ 0. Let usnow show that dimH

ℓ(N) is bounded from above by a constant depending onlyon n and K. We fix p-harmonic ℓ-forms ξi generating Hℓ(N), with p = n/ℓ. Thiscan be done by a result of Scott [42]. Furthermore, we may assume that formsξi are uniformly separated and uniformly bounded in Lp, that is, ‖ξi − ξj‖p ≥ 1and ‖ξi‖p = 1 for every i and j.

A local version [34, Theorem 6] of the value distribution result of Mattila andRickman yields that

(8.6)

∫

Bn(r)\Bn(2)

|f ∗ξ|n/ℓ ∼

∫

Bn(r)\Bn(2)

Jf

for large radii r. Using Theorem 8.3 and a decomposition technique due toRickman, we find a radius R and a decomposition of the annulus Bn(R) \Bn(2)


into domains quasiconformally equivalent to Bn in such a way that we have a

quasiregular embedding ψ : Bn → Rn \ Bn(2) with properties

(8.7)

∫

ψBn(1/2)

Jf &

(∫

Bn(R)\Bn(2)

Jf

)1/4

and

(8.8)

∫

ψBn

Jf .

∫

Bn(R)\Bn(2)

Jf .

Combining (8.6) with (8.7) and (8.8), we have that forms ϕ∗f∗ξi are uniformly

bounded in Lp(Bn) and uniformly separated in L

p(Bn(1/2)). By compactness,the number of forms is bounded by a constant depending on data.

Remark 8.4. The use of A-harmonic forms in the proof of Theorem 8.2 is verysimilar to their use in the proof of Theorem 8.1. Also Theorem 8.3 correspondsto a theorem of Bonk and Heinonen ([4, Theorem 1.11]).

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Ilkka Holopainen Address: Department of Mathematics and Statistics, P.O. Box 68,

FIN-00014 University of Helsinki, Finland

E-mail: [email protected]

Pekka Pankka Address: Department of Mathematics and Statistics, P.O. Box 68,

FIN-00014 University of Helsinki, Finland



Hyperbolic-type metrics

Henri Linden

Abstract. The article is a status report on the contemporary research ofhyperbolic-type metrics, and considers progress in the study of the classesof isometry- and bilipschitz mappings with respect to some of the presentedmetrics. Also, the Gromov hyperbolicity question is discussed.

Keywords. Hyperbolic-type metric, intrinsic metric, isometry problem, bilipschitz-mapping, Gromov hyperbolic space.

2000 MSC. Primary 30F45, Secondary 30C65,53C23.

Contents

1. Introduction 151

2. The metrics 152

3. Isometries and bilipschitz-mappings 157

4. Gromov hyperbolicity 160

References 163

1. Introduction

In geometric function theory there are many different distance functions around,which — to a greater or lesser degree — resemble the classical hyperbolic metric.Some of these are defined by geometric means, some by implicit formulas, andmany by integrating over certain weight functions.

What all these metrics have in common, is that they are defined in someproper subdomain D ( R

n, and are strongly affected by the geometry of the do-main boundary. Thus we should actually speak of families of metrics dDD(Rn ,since the metric looks different in each domain, even though the defining formulamight be the same. In the literature, however, one usually abuses notation andspeaks only of “the metric d”, which we will do here also. The metrics typicallyhave negative curvature, ie. the geodesics, if they exist, avoid the boundary.Most of the metrics described here also have an invariance property in the sensethat

dD(x, y) = df(D)(f(x), f(y)),(1.1)

152 H. Linden IWQCMA05

for mappings f belonging to some fixed class, say similarities, Mobius transfor-mations, or conformal mappings.

Many of the metrics, especially those with simple explicit formulas, have beendeveloped as tools for estimating other, more hard-to-handle metrics, such as thequasihyperbolic metric, which is probably the one most commonly used metricpresented in this text. It has found applications in many branches of analysis,and is a very natural generalization of the classical hyperbolic metric to anydomain D and dimension n ≥ 2. It has some flaws though, in most cases onecannot compute it, and actually very little is known about the metric itself. Thedifficulty of explicit computation is typical also for some other metrics, and forthis reason we have a lot of “similar” metrics around, which in many cases areequivalent to each other; a handy feature, if one metric is suited for your study,but the other is not. Here we will try to give a survey on some of these metrics.

2. The metrics

The classical starting point is the hyperbolic geometry developed by Poincareand Lobachevsky in the early 19:th century. Poincare used the unit ball asdomain for his model, and Lobachevsky used the half space. These modelsturned out to be equivalent in the sense that Mobius transformations betweenthem are isometries.

2.1. Definition. Let D ∈ Hn,Bn, and define a weight (or density) function

w : D → R by

w(z) =1

dist(z, ∂D), for D = Hn and w(z) =

2

1 − |z|2 , for D = Bn.

Then the hyperbolic length ℓρ(γ) of a curve γ is defined by

ℓρ(γ) = ℓρ,D(γ) =

∫

γ

w(z) |dz|,(2.2)

where |dz| denotes the length element. After this, the hyperbolic distance ρD isdefined for all x, y ∈ D by

ρD(x, y) = infγ∈Γxy

ℓρ,D(γ) = infγ∈Γxy

∫

γ

w(z) |dz|,(2.3)

where Γxy is the family of all rectifiable curves joining x and y within D.

The above method to define metrics is frequently used. In fact, to get acompletely new metric, the only thing that needs to be changed is the weightfunction. After that, the length and the new distance function are defined as in(2.2) and (2.3), respectively. The benefit of defining a metric d like this is thatit will automatically be intrinsic, in other words, it will be its own inner metric

d. This means that

dD(x, y) = dD(x, y) := infγ∈Γxy

ℓd,D(γ).(2.4)

Hyperbolic-type metrics 153

2.5. Geodesics. When a metric is defined in the way described above, onemight ask how to find the curve γ ∈ Γxy giving the desired infimum (which —if it is found — is in fact a minimum). In general, this can be far from trivial,even if such a curve exists. Curves minimizing the distance in this way arecalled geodesics or geodesic segments. Another way of characterizing a geodesic,is that it satisfies the triangle inequality with equality, ie. the curve γ ∈ Γxy is ageodesic, if for all u, v, w ∈ |γ| properly ordered, we have

dD(u,w) = dD(u, v) + dD(v, w).

We denote by JdD[x, y] the geodesic segment between x and y in (D, d). This

segment may, however, not be unique, and no particular choice is made here. Ametric space in which geodesic segments exist between any two given points, iscalled a geodesic metric space. If, in addition, the geodesic is unique, the spaceis totally geodesic. Naturally a geodesic metric is always intrinsic.

2.6. Hyperbolic metric in G. It is also possible to define the hyperbolicmetric in a general simply connected subdomain G of the plane, since by theRiemann mapping theorem there exists a conformal mapping f : G → fG = B2.Then the metric density is defined by

ρG(z) = ρB2(f(z))|f ′(z)|.

From the Schwarz lemma it follows that ρG is independent of the choice of f .We then define the hyperbolic metric hG by (2.3) using the density ρG. Thisdefinition automatically gives the hyperbolic metric the invariance property of(1.1) for the class of conformal mappings. Note, that while in the classical caseswe use the traditional notation ρBn and ρHn for the hyperbolic metric, in generaldomains we use hG. Also, note that when the dimension n ≥ 3, every conformalmapping is a Mobius mapping, so it is not possible to extend the definition togeneral simply connected domains like above. In fact, for n ≥ 3 the hyperbolicmetric is defined only in Bn and Hn.

The hyperbolic metric is well understood, and the geodesic flow is known. Infact, in the classical models Bn and Hn the geodesics are known to be circulararcs orthogonal to the boundary, and in other domains the geodesics simply areinduced by the conformal mapping. Moreover, for the classical cases there areexplicit formulas to calculate the value of the hyperbolic metrics in terms ofeuclidean distances. For a comprehensive study on the classical cases, see thebook by Beardon [Be1]. The hyperbolic metric in an arbitrary domain has beenstudied by F. Gehring, K. Hag and A. Beardon, see eg. the articles [Be3] and[GeHa1].


Bn

x

y x

y

Hn

x

y

x y* *

*

*

Figure 1: Hyperbolic geodesics in Bn and Hn.

One way to calculate the hyperbolic distance, is to use the absolute cross-ratio

defined by

|a, b, c, d| =|a − c||b − d||a − b||c − d| , a, b, c, d ∈ R

n.

One can prove that, if C is the circle containing JρBn [x, y] or JρHn [x, y] andx∗

, y∗ = C ∩∂Bn or x∗

, y∗ = C ∩∂Hn in the same order as in Figure 1, then

ρBn(x, y) = log |x∗, x, y, y

∗| = ρHn(x, y).(2.7)

Other explicit formulas have also been derived, see the book [Be1].

2.8. The Apollonian metric. The formula in (2.7) makes one wonder whethera similar approach could be generalized to any domain D ( R

n. It turns outthat this is very much possible; the Apollonian distance in a domain D is definedby

αD(x, y) = supz,w∈∂D

log|z − x||z − y|

|w − y||w − x| ,(2.9)

for all x, y ∈ D. This is a metric, unless the boundary is the subset of a circleor a line, in which case it is only a pseudo-metric, ie. the metric axiom d(x, y) =0 ⇒ x = y need not hold.

Geometrically the Apollonian metric can be thought of in the following way:an Apollonian circle (or sphere, when n ≥ 3) with respect to the pair (x, y), is aset

Bx,y,q =

z ∈ R

n

∣∣∣∣|z − x||z − y| = q

.

Then the Apollonian metric is

αD(x, y) = log qxqy,

where qx and qy are the ratios of the largest possible balls Bx,y,qxand By,x,qy

stillcontained in D.


z

w

D

B

By

xx

y

Figure 2: The Apollonian balls approach.

The Apollonian metric is invariant in Mobius mappings in the sense of (1.1).It is an easy exercise in geometry to show that in the case D = Hn the points z

and w are actually the points x∗ and y

∗ in (2.7), and thus ρHn = αHn .

The Apollonian metric has been studied in [GeHa2] and [Se], but especiallyby P. Hasto and Z. Ibragimov in a series of articles, see e.g. [Ha1],[Ha2],[HaIb]and [Ib].

The Apollonian metric is in a way a convenient construction with a cleargeometric interpretation, but as a shortcoming it has its lack of geodesics. In thearticle [HaLi] some work is done to overcome this problem, by introducing thehalf-Apollonian metric, defined by

(2.10) ηD(x, y) = supz∈∂D

∣∣∣∣log|x − z||y − z|

∣∣∣∣ ,

for all x, y ∈ D. The geometric intuition here is the same as for the Apollonianmetric, Indeed, instead of log qxqy we have

ηD(x, y) = log maxqx, qy.This metric is a only similarity invariant, but instead it has more geodesics thanthe Apollonian metric. It is also bilipschitz equivalent to the Apollonian metric,in fact

1

2αD(x, y) ≤ ηD(x, y) ≤ αD(x, y).

2.11. The quasihyperbolic metric. The quasihyperbolic metric is perhapsthe most well-known and frequently used of the metrics considered here. It wasdeveloped by F. Gehring and his collaborators in the 70’s. It is defined by themethod of 2.1 using

w(z) =1

dist(z, ∂D), z ∈ D

as weight function. It is immediate that for D = Hn the quasihyperbolic metriccoincides with the hyperbolic metric ρHn . The quasihyperbolic metric is invariantunder the class of similarity mappings.


The quasihyperbolic metric is well-behaved in many senses: the weight func-tion is quite simple and it is a natural generalization of the hyperbolic metric.Also, it is known to be geodesic for any domain D ( R

n [GeOs]. One of theshortcomings of the metric is that in general the geodesics are not easy to deter-mine. Besides the half-space Hn, the geodesics are known in the punctured spaceR

n \z and in the ball Bn, see [MaOs]. Recently the geodesics were determinedalso for the punctured ball Bn \ 0, and planar angular domains

Sϕ = (r, θ) | 0 < θ < ϕ, 0 < ϕ < 2π,

see [Li1].

2.12. Distance-ratio metrics. As the quasihyperbolic metric cannot be ex-plicitly evaluated in the case of general domains, a typical way to overcome thisproblem is to approximate it by another metric, often one of the distance-ratio

metrics or j-metrics. (Actually, by their construction also the Apollonian andhalf-Apollonian metrics could be described as “distance-ratio metrics”). Thereare two versions of these. The first, introduced by F. Gehring, is defined by

(2.13) jD(x, y) = log

(1 +

|x − y|dist(x, ∂D)

) (1 +

|x − y|dist(y, ∂D)

), x, y,∈ D.

The other one is defined by

(2.14) jD(x, y) = log

(1 +

|x − y|dist(x, ∂D) ∧ dist(y, ∂D)

). x, y,∈ D.

is a modification due to M. Vuorinen.

The two metrics have much in common, but also important differences, whichwill be discussed further in Sections 2 and 3. Both are similarity invariant, andcan be used to estimate the quasihyperbolic metric. The metrics satisfy therelation

jD(x, y) ≤ jD(x, y) ≤ 2 jD(x, y), x, y ∈ D.

The lower bound for the quasihyperbolic metric is given by the inequality

jD(x, y) ≤ kD(x, y)

proved in [GePa], which holds for points x, y in any proper subdomain D. Theupper bound holds for so called uniform domains, which is a wide class of domainsintroduced in [MaSa].

2.15. Definition. A domain D ( Rn is called uniform or A-uniform, if there

exists a number A ≥ 1 such that the inequality

kD(x, y) ≤ A jD(x, y)

holds for all x, y ∈ D.

There are many definitions for uniform domains around, see eg. [Ge], so oftenmany “nice” domains can be shown to be uniform by other means, and so onehas access to the inequality in 2.15. However, typically very little can be saidabout the constant A. These matters have been studied in [Li1].


The j-metric defined in (2.14) has another important connection to the quasi-hyperbolic metric. The quasihyperbolic metric is namely the inner metric of thej-metric, in the sense of (2.4). In other words

kD(x, y) = infγ∈Γxy

ℓj,D(γ).

Since the j-metric fails to be intrinsic, it cannot be geodesic either. In fact,the j-metric has geodesics only in some special cases, see [HaIbLi, 3.7]. Very

little is known about the geodesic segments of the j-metric, although it can beconjectured that there is not much of them either.

3. Isometries and bilipschitz-mappings

As pointed out earlier, most of the hyperbolic-type metrics defined in thisarticle satisfy some kind of invariance property, that is, they satisfy the equality(1.1) for some class of mappings f . Typically this invariance property followsalmost directly from the definition of the metric, for instance, it is easy to seefrom the formulas (2.9) and (2.10) that the Apollonian metric is Mobius-invariantand the half-Apollonian metric is similarity invariant. The interesting questionmostly regards the other implication. Is the class of “natural candidates” theonly mappings which give isometries in the metric in question? And what are the“near-isometries”, that is, the bilipschitz mappings? There are still many openends regarding these questions, though some progress has been made recently.

3.1. Definition. Let D and D′ = f(D) be domains such that equipped with

distances dD and dD′ they are metric spaces. Then a continuous mapping f : D →D

′ is said to be L-bilipschitz in (or with respect to) the metric d if for all x, y ∈ D

we have1

LdD(x, y) ≤ dD′(f(x), f(y)) ≤ L dD(x, y)

for some L ≥ 1. If the above inequality holds with L = 1, f is a d-isometry.

3.2. “One-point” and “two-point” metrics. In general, the hyperbolic-type metrics can be divided into length-metrics, defined by means of integratinga weight function, and point-distance metrics. The point-distance metrics mayagain be classified by the number of boundary points used in their definition.So for instance the j-metric and the half-Apollonian metric would be ‘one-pointmetrics”, whereas the j, and the Apollonian metrics are “two-point metrics”.

Actually also the length metrics can be characterized in the same way, bylooking at their weight function. Then the quasihyperbolic metric is a one-pointmetric. An example of a two-point length metric is the so called Ferrand metric

σD, see [Fe1]. It is defined for a domain D ( Rn

with card ∂D ≥ 2, using theweight function

wD(x) = supa,b∈∂D

|a − b||x − a||x − b| , x ∈ D \ ∞.


This metric is Mobius invariant and coincides with the hyperbolic metric on Hn

and Bn. Moreover, it is bilipschitz equivalent to the quasihyperbolic metric bythe inequality

(3.3) kD(x, y) ≤ σD(x, y) ≤ 2 kD(x, y), x, y ∈ D.

Naturally one would expect the one-point point-distance metrics to be theeasiest ones to study. In fact, much can be said about these metrics when itcomes to the isometry question. The half-Apollonian metric has recently beenstudied in [HaLi]. A point x ∈ D is called circularly accessible if there existsa ball B ⊂ G such that x ∈ ∂B. If x is circularly accessible by two distinctballs whose surfaces intersect at more than one point, it is called a corner point,otherwise a regular point.

3.4. Theorem. Let D ( Rn be a domain which has at least n regular boundary

points which span a hyperplane. Then f : D → Rn is a homeomorphic η-isometry

if and only if it is a similarity mapping.

Furthermore, it was shown that Mobius mappings are in fact 2-bilipschitzwith respect to ηD.

For the j-metric, some results can be found in [HaIbLi], and in fact in aslightly more general setting. The implications for the j-metric can be expressedas follows;

3.5. Corollary. Let D ( Rn. Then f : D → R

n is a j-isometry if and only if

(1) f is a similarity, or

(2) D = Rn \ a and, up to similarity, f is the inversion in a sphere centered

at a.

Since jD = kD, it immediately follows that every isometry of the j-metric isan isometry of the quasihyperbolic metric, of course in this case that does notprovide us with very much new information. However, a similar relation is truefor the Seittenranta metric δD defined in [Se] by

δD(x, y) = log

(1 + sup

a,b∈∂D

|x − y||a − b||a − x||b − y|

), x, y ∈ D,

which is also studied in [HaIbLi]. Namely, here we have that δD = σD, so wedirectly see that this is a Mobius invariant metric. In [Se] it is proved that at leastEuclidean bilipschitz mappings are bilipschitz with respect to δ. The converse isnot true, as can be shown by the counterexample

f : B2 \ 0 → B2 \ 0, f(x) = |x| · x.

However, in [Se] it was shown that every bilipschitz δ-mapping is a quasicon-formal mapping, and that every δ-isometry is conformal with respect to theEuclidean metric (and thus Mobius for n ≥ 3). In [HaIbLi] it was shown thatalso for n = 2 in fact the δ-isometries are exactly the Mobius mappings.


For the j-metric there are still many open problems regarding the bilipschitzquestion. It is well known (see [Vu]), that an Euclidean L-bilipschitz mapping isL

2-bilipschitz with respect to the j (and k) metric.

For the Apollonian metric the isometry and bilipschitz questions have beenstudied by several authors. The work was started by Beardon in [Be2], and con-tinued by Gehring and Hag in [GeHa2] where they studied Apollonian bilipschitzmappings. They proved the following theorem.

3.6. Theorem. Let D ( R2 be a quasidisk and f : D → D

′ be an Apollonian

bilipschitz mapping.

(1) If D′ is a quasidisk, then f is quasiconformal in D and f = g|D, where

g : R2 → R

2

is quasiconformal.

(2) If f is quasiconformal in D, then D′ is a quasidisk and f = g|D, where

g : R2 → R

2

is quasiconformal.

In [Ha2] the above property (1) was generalized to hold also for n ≥ 3. In thesame article also a condition was introduced which determines when a Euclideanbilipschitz mapping is also Apollonian bilipschitz. In the article [HaIb] it isshown that for n = 2 the Apollonian isometries are exactly restrictions of Mobiusmappings.

For the quasihyperbolic metric the question regarding the isometries has longbeen open. In [MaOs] it was shown that every kD-isometry is a conformal map-ping. A similar proof gives the same result for Ferrand’s metric σD. However, in[Ha3] it is shown that if the boundary of the domain is regular enough (C3, or C

2

unless the domain is either strictly convex or has strictly convex complement),then the quasihyperbolic isometries are exactly the similarity mappings.

3.7. Conformal modulus. We conclude by introducing two new metrics whichare particularly interesting regarding the question of bilipschitz mappings. Let Γbe a family of curves in R

n. By F(Γ) we denote the family of admissible functions,

that is, non-negative Borel-measurable functions ρ : Rn → R such that

∫

γ

ρ ds ≥ 1

for each locally rectifiable curve γ ∈ Γ. The n-modulus or the conformal modulus

of Γ is defined by

M(Γ) = Mn(Γ) = infρ∈F(Γ)

∫

Rn

ρn

dm,

where m is the n-dimensional Lebesgue measure. It is a conformal invariant,i.e. if f : G → G

′ is a conformal mapping and Γ is a curve family in G, thenM(Γ) = M(fΓ).

For E,F,G ⊂ Rn

we denote by ∆(E,F ; D) the family of all closed non-constant curves joining E and F in D, that is, γ : [a, b] → R

nbelongs to

∆(E,F ; D) if one of γ(a), γ(b) belongs to E and the other to F , and furthermoreγ(t) ∈ D for all a < t < b.


Now we will define two new conformal invariants in the following way. Forx, y ∈ D ( R

nλD is defined by

λD(x, y) = infCx,Cy

M(∆(Cx, Cy; D)

),

where Cz = γz[0, 1) and γz : [0, 1] → D is a curve such that z ∈ |γz| and γz(t) →∂D when t → 1 and z = x, y. Correspondingly,

µD(x, y) = infCxy

M(∆(Cxy, ∂D; D)

),

where Cxy is such that Cxy = γ[0, 1] and γ is a curve with γ(0) = x and γ(1) = y.

It is not difficult to show that both quantities µD and λD are conformal invari-ants, and that µD is a metric (often called the modulus metric) when cap ∂D > 0,

see [Ga]. λD is not a metric, but λ∗D = λ

1/(1−n)

D introduced in [Fe2] is, as long asthe boundary of the domain has more then two points.

One of the interesting feature regarding these metrics is that both are easilyseen — by their definitions — to be conformal invariants. Moreover, the followingcan be shown (see [Vu, 10.19]);

3.8. Theorem. If f : D → D′ = fD is a quasiconformal mapping, then

(1) µD(x, y)/L ≤ µfD(f(x), f(y) ≤ L µD(x, y),(2) λ

∗D(x, y)/L1/(n−1) ≤ λ

∗fD(f(x), f(y)) ≤ L

1/(n−1)λ∗D(x, y)

hold for all x, y ∈ D, where L = maxKI(f), KO(f) is the maximal dilatation

of f .

It is not known if the class of bilipschitz mappings with respect to µ or λ∗

includes any other than quasiconformal mappings.

4. Gromov hyperbolicity

One way of telling “how hyperbolic” a metric in fact is, is to study whetherit satisfies hyperbolicity in the sense of M. Gromov. Classically such spaceshave been studied in the geodesic case, and then a space is said to be Gromov

δ-hyperbolic if for all triples of geodesics Jd[x, y], Jd[y, z] and Jd[x, z] we havethat

dist(w, Jd[y, z] ∪ Jd[z, x]) ≤ δ

for all w ∈ Jd[x, y], i.e. if all geodesic triangles are δ-thin.

4.1. The Gromov product. In non-geodesic spaces, however, we are con-strained to use the definition involving the Gromov product. This can be definedfor two points x, y ∈ D with respect to a base point w by setting

(x|y)w =1

2

(d(x,w) + d(y, w) − d(x, y)

).


A space is then said to be Gromov δ-hyperbolic if it satisfies the inequality

(x|z)w ≥ (x|y)w ∧ (y|z)w − δ

for all x, y, z ∈ D and a base point w ∈ D. A space is said to be Gromov

hyperbolic if it is Gromov δ-hyperbolic for some δ. Sometimes one wants to usethe equivalent definition for Gromov hyperbolicity

d(x, z) + d(y, w) ≤(d(x,w) + d(y, z) ∨ d(x, y) + d(z, w)

)+ 2δ.(4.2)

Recently the study of Gromov hyperbolicity has become quite popular, andeven hyperbolicity results on particular metrics in geometric function theoryhave been developed by a number of authors. A systematic study of the differentmetrics is made easier by the fact that Gromov hyperbolicity is preserved bycertain classes of mappings, so called rough isometries. We say that two metricsd and d

′ are roughly isometric if there exists a positive constant C such that

d(x, y) − C ≤ d′(x, y) ≤ d(x, y) + C.

It is immediately clear from the definition (4.2) that roughly isometric metricsare Gromov hyperbolic in the same domains. Moreover, we say that two metricsare (A,C)-quasi-isometric if there is A ≥ 1, C ≥ 0 such that

A−1

d(x, y) − C ≤ d′(x, y) ≤ A d(x, y) + C.

Also quasi-isometries (and thus bilipschitz mappings) are known to preserve Gro-mov hyperbolicity, provided that the spaces are geodesic.

Naturally we would want the hyperbolic metric itself to be Gromov hyperbolicalso, and in fact it is, with constant δ = log 3, as is shown in [CoDePa]. One ofthe more interesting and general results is one from the comprehensive study ofM. Bonk, J. Heinonen and P. Koskela [BoHeKo], where it is shown that for auniform domain D the space (D, kD) is always Gromov hyperbolic.

For many of the other metrics Gromov hyperbolicity is easily proved or dis-proved using the results from [Ha4]. Namely, it turns out that the j-metric isGromov hyperbolic in every proper subdomain of R

n, whereas the j-metric isGromov hyperbolic only in R

n \ a. Then, using inequalities

jD(x, y) − log 3 ≤ ηD(x, y) ≤ jD(x, y),

jD(x, y) − log 9 ≤ αD(x, y) ≤ jD(x, y),

and

αD(x, y) ≤ δD(x, y) ≤ αD(x, y) + log 3

we immediately get some results by rough isometry, that is, the results in Table1 regarding the Apollonian, half-Apollonian and Seittenranta metrics. For prov-ing Gromov hyperbolicity of the Ferrand metric one can use geodesity, Gromovhyperbolicity of the quasihyperbolic metric, and the bilipschitz equivalence in(3.3).

Finally, for the µ and λ∗ metrics positive results regarding Gromov hyperbol-icity are shown in [Li2].


4.3. Theorem. The metric space (Bn, λ

∗Bn) is Gromov δ-hyperbolic, with Gro-

mov constant

δ ≤ 1

2

(ωn−1

2

) 1

1−n

(log 64

3+ 4 log λn

)≤ 1

2

(ωn−1

2

) 1

1−n

(log 64

3+ 4(log 2 + n − 1)

),

where ωn−1 denotes the (n − 1)-dimensional surface area of Sn−1 and λn is the

Grotzsch constant. Also, any simply connected proper subdomain D ( R2 is

Gromov δ-hyperbolic with respect to the metric λ∗G, where

δ ≤ log 5462

2π≈ 1.3696.

4.4. Theorem. The metric space (Bn, µBn) is Gromov δ-hyperbolic, with Gro-

mov constant

δ ≤ 2n−1cn log 12,

where cn is the spherical cap inequality constant, see [Vu]. Especially, every

simply connected domain D ( R2 is Gromov hyperbolic with

δ ≤ 2 log 12

π≈ 1.5819.

4.5. Theorem. The metric space (Rn \ z, λ∗Rn\z) is Gromov hyperbolic, with

δ ≤ 2ω1

n−1

n−1log 18λ2

n ≤ 2ω1

n−1

n−1

(log 72 + 2n − 2

).

As the below table indicates, the j-metric and the half-Apollonian metric arethe only metrics of the ones discussed here which fail to be Gromov hyperbolicin most cases. These results indicate that these metrics are in a way “too easy”,or have too little structure for satisfying Gromov hyperbolicity. On the otherhand, in other contexts that is one of their strongest features, as has been seenin earlier sections.

Domain condition Proved where

kD D uniform [BoHeKo]hD n = 2 all domains defined, n ≥ 3, D = Bn,Hn [CoDePa] and conf. invarianceαD All domains D ( R

n [Ha4] and rough isometryηD Only D = R

n \ z, δ = log 9 [Ha4],[HaLi]jD Only D = R

n \ z, δ = log 9 [Ha4]

jD All domains D ( Rn [Ha4]

δD All domains D ( Rn [Ha4],[Se]

σD D uniform, for D = Bn δ = log 3 [Fe1],[BoHeKo]λ∗

D D = Bn, Rn∗ , n = 2 simply conn. domains [Li2]

µD D = Bn, n = 2 simply conn. domains [Li2]

Table 1: Gromov hyperbolicity of some metrics.


References

[Be1] A. F. Beardon: The geometry of discrete groups. Graduate Texts in Mathe-matics, Vol. 91, Springer-Verlag, Berlin-Heidelberg-New York, 1982.

[Be2] A. F. Beardon: The Apollonian metric of a domain in Rn. Quasiconformal

mappings and analysis (Ann Arbor, Michigan, 1995), Springer-Verlag, NewYork, (1998), 91–108.

[Be3] A. F. Beardon: The hyperbolic metric in a rectangle II. Ann. Acad. Sci. Fenn.Math., 28, (2003), 143–152.

[BoHeKo] M. Bonk, J. Heinonen and P. Koskela: Uniformizing Gromov hyperbolic

spaces. Asterisque 270, 2001, 1–99.[CoDePa] M. Coornaert, T. Delzant and A. Papadopoulos: Geometrie et theorie

des groupes. Lecture Notes in Mathematics, Vol. 1441 Springer-Verlag, Berlin,1990. (French, english summary).

[Fe1] J. Ferrand: A characterization of quasiconformal mappings by the behavior of

a function of three points. Proceedings of the 13th Rolf Nevanlinna Colloquium(Joensuu, 1987; I. Laine, S. Rickman and T. Sorvali (eds.)), Lecture Notes inMathematics Vol. 1351, Springer-Verlag, New York, (1988), 110–123.

[Fe2] J. Ferrand: Conformal capacity and extremal metrics. Pacific J. Math. 180,no. 1, (1997), 41–49.

[Ga] I. S. Gal: Conformally invariant metrics and uniform structures. Indag. Math.22, (1960), 218–244.

[Ge] F. W. Gehring: Characteristic properties of quasidisks. Les Presses del’Universite de Montreal, Montreal, 1982.

[GeHa1] F. W. Gehring and K. Hag: A bound for hyperbolic distance in a quasidisk.

Computational methods and function theory (Nicosia, 1997), 233–240, Ser. Ap-prox. Decompos. 11, World Sci. Publishing, River Edge, NJ, 1999.

[GeHa2] F. W. Gehring and K. Hag: The Apollonian metric and quasiconformal

mappings. In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998), 143–163, Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 2000.

[GeOs] F. W. Gehring and B. G. Osgood: Uniform domains and the quasi-

hyperbolic metric. J. Anal. Math. 36 (1979), 50–74.[GePa] F. W. Gehring and B. Palka: Quasiconformally homogeneous domains. J.

Anal. Math. 30 (1976), 172–199.[Ha1] P. Hasto: The Apollonian metric: uniformity and quasiconvexity. Ann. Acad.

Sci. Fenn. Math., 28, (2003), 385–414.[Ha2] P. Hasto: The Apollonian metric: limits of the approximation and bilipschitz

properties. Abstr. Appl. Anal., 20, (2003), 1141–1158.[Ha3] P. Hasto: Isometries of the quasihyperbolic metric. In preparation (2005).

Available at http://www.helsinki.fi/˜hasto/pp/

[Ha4] P. Hasto: Gromov hyperbolicity of the jG and jG metrics. Proc. Amer. Math.Soc. 134, (2006), 1137–1142.

[HaIb] P. Hasto and Z. Ibragimov: Apollonian isometries of planar domains are

Mobius mappings. J. Geom. Anal. 15, no. 2, (2005), 229–237.[HaIbLi] P. Hasto, Z. Ibragimov and H. Linden: Isometries of relative metrics. In

preparation (2004). Available at http://www.helsinki.fi/˜hlinden/pp.html

[HaLi] P. Hasto and H. Linden: Isometries of the half-Apollonian metric. Compl.Var. Theory Appl. 49, no. 6 (2004), 405–415.

[Ib] Z. Ibragimov: On the Apollonian metric of domains in Rn

Compl. Var. TheoryAppl. 48, no. 10, (2003), 837–855.

[Li1] H. Linden: Quasihyperbolic geodesics and uniformity in elementary domains.

Ann. Acad. Sci. Fenn. Math. Diss. 146, (2005), 1–52.


[Li2] H. Linden: Gromov hyperbolicity of certain invariant metrics. In preparation.Available as preprint in Reports Dept. Math. Stat. Univ. Helsinki 409, (2005),University of Helsinki.

[MaOs] G. Martin and B. Osgood: The quasihyperbolic metric and the associated

estimates on the hyperbolic metric. J. Anal. Math. 47 (1986), 37–53.[MaSa] O. Martio and J. Sarvas: Injectivity theorems in plane and space. Ann.

Acad. Sci. Fenn. Ser. A I Math. 4, (1978/79), 383–401.[Se] P. Seittenranta: Mobius-invariant metrics. Math. Proc. Camb. Phil. Soc.

125 (1999), 511–533.[Vu] M. Vuorinen: Conformal geometry and quasiregular mappings. Lecture Notes

in Mathematics, Vol. 1319 Springer-Verlag, Berlin, 1988.

Henri Linden E-mail: [email protected]: P.O.Box 68, 00014 University of Helsinki, FINLAND


Geometric properties of hyperbolic geodesics

W. Ma and D. Minda

Abstract. In the unit disk D hyperbolic geodesic rays emanating from theorigin and hyperbolic disks centered at the origin exhibit simple geometricproperties. The goal is to determine whether analogs of these geometric prop-erties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simplyconnected region Ω. According to whether the simply connected region Ω is asubset of the unit disk D, the complex plane C or the extended complex plane(Riemann sphere) C∞ = C∪ ∞, the geometric properties are measured rel-ative to the background geometry on Ω inherited as a subset of one of theseclassical geometries, hyperbolic, Euclidean and spherical. In a simply con-nected hyperbolic region Ω ⊂ C hyperbolic polar coordinates possess globalEuclidean properties similar to those of hyperbolic polar coordinates aboutthe origin in the unit disk if and only if the region is Euclidean convex. Forexample, the Euclidean distance between travelers moving at unit hyperbolicspeed along distinct hyperbolic geodesic rays emanating from an arbitrarycommon initial point is increasing if and only if the region is convex. A simpleconsequence of this is the fact that the two ends of a hyperbolic geodesic in aconvex region cannot be too close. Exact analogs of this Euclidean separatingproperty of hyperbolic geodesic rays hold when Ω lies in either the hyperbolicplane D or the spherical plane C∞.

Keywords. hyperbolic metric, hyperbolic geodesics, hyperbolic disks, Eu-clidean convexity, hyperbolic convexity, spherical convexity.

2000 MSC. Primary 30F45; Secondary 30C55.

Contents

1. Introduction 166

2. Hyperbolic polar coordinates in the unit disk 167

3. Hyperbolic polar coordinates in a disk or half-plane 169

4. Hyperbolic polar coordinates in simply connected regions 172

5. Euclidean convex univalent functions 173

6. Euclidean convex regions 175

7. Spherical geometry 177

8. Spherically convex univalent functions 179

9. Spherically convex regions 182

Version October 19, 2006.The second author was supported by a Taft Faculty Fellowship.

166 Ma and Minda IWQCMA05

10. Hyperbolic geometry 183

11. Concluding remarks 185

References 186

1. Introduction

The results in this expository paper are adapted from [16] and [17] and concerngeometric properties of hyperbolic geodesics in a simply connected hyperbolicregion Ω and, to a lesser extent, geometric properties of hyperbolic disks. Thesetwo references contain many results not mentioned here and as well as the detailsthat are not presented in this largely expository article. In particular, proofs notgiven in this article can be found in these two references. There are three differentcases to consider according to whether the region Ω is a subset of the hyperbolicplane D, the Euclidean plane C, or the spherical plane C∞ = C ∪ ∞. Twogeometries on the region Ω will be considered. First, the intrinsic hyperbolicgeometry on Ω and, second, the geometry that Ω inherits as a subset of thehyperbolic, Euclidean or spherical plane.

Here is a rough description of the types of behavior of hyperbolic geodesicsthat we will consider. Fix a point w0 ∈ Ω. For θ ∈ R, let ρ(w0, Ω) denote thehyperbolic geodesic ray emanating from w0 that has unit Euclidean tangent e

iθ

at w0 and let w0(s, θ) be the hyperbolic arc length parametrization of this geo-desic. Under what conditions does the point w0(s, θ) move monotonically awayfrom w0 when s increases? Here motion away from w0 is measured relative tothe background distance. For example, if Ω lies in the Euclidean plane, thismeans the Euclidean distance |w0(s, θ)−w0| should increase with s. The secondtype of behavior we consider is whether the background distance between dis-tinct geodesic rays increases as points move along these rays. In the Euclideancase we inquire whether the Euclidean distance |w0(s, θ1) − w0(s, θ2)| increaseswith s when e

iθ1 6= eiθ2 . Intuitively, one can think of two travelers departing

from w0 at the same time along different hyperbolic geodesic rays and travelingat unit hyperbolic speed along the geodesics and asking whether the travelersseparate monotonically in the Euclidean sense. Finally, we investigate the shapeof hyperbolic circles relative to the background geometry. The main concern iswhether hyperbolic circles are convex curves relative to the background geome-try. Hyperbolic rays emanating from a point w0 together with hyperbolic circlescentered at w0 form the coordinate grid for hyperbolic polar coordinates in Ω, soour work can be interpreted as studying geometric properties of the hyperbolicpolar coordinate grid relative to the background geometry.

A descriptive outline of the paper follows. Hyperbolic polar coordinates inthe unit disk are defined in Section 2, while Section 3 extends hyperbolic polarcoordinates to any Euclidean disk or half-plane. Simple Euclidean properties ofthe hyperbolic polar coordinate grid in any disk or half-plane are established asthe model for future investigations. Hyperbolic polar coordinates for a simply

Geometric properties of hyperbolic geodesics 167

connected region are introduced in Section 4. Loosely speaking hyperbolic polarcoordinates can be transferred from the unit disk to a simply connected regionΩ by using the Riemann Mapping Theorem; a conformal map f : D → Ω is ahyperbolic isometry. Characterizations of Euclidean convex univalent functionsare discussed in Section 5. in Section 6 these characterizations are used to es-tablish Euclidean properties of hyperbolic polar coordinates in Euclidean convexregions and to show that these properties characterize Euclidean convex regions.The remainder of the paper is devoted to analogs of these results in the sphericaland hyperbolic planes. The spherical plane is introduced in Section 7 along withthe notion of a spherically convex region. The results for regions in the sphericalplane parallels the Euclidean context. Spherically convex univalent functions arepresented in Section 8. The reader should note the number of parallels betweenspherically convex univalent functions and Euclidean convex univalent functions.The results for spherically convex univalent functions seem more involved thanthose for Euclidean convex univalent functions; the more complicated nature offormulas relating to spherically convex univalent functions is due to the fact thatthe spherical metric has curvature 1 while the Euclidean metric has curvature0. Nonzero curvature causes the appearance of extra terms. Applications ofsome results for spherically convex functions to the behavior of the hyperboliccoordinate grid in a spherically convex region are given in Section 9. Section 10considers the behavior of the hyperbolic polar coordinate grid for hyperbolicallyconvex regions in the unit disk. Because of the strong similarity with the previ-ous cases for Euclidean convexity and spherical convexity, we present a concisediscussion of the results. The reader should note that some theorems for hy-perbolically convex univalent functions formally differ from those for sphericallyconvex univalent functions by certain sign changes; these alterations in sign aredue to the fact that the hyperbolic plane has curvature −1 while the sphericalplane has curvature 1. The brief final section directs the reader to some othersituations in function theory in which there are parallel results for the hyperbolic,Euclidean and spherical planes.

2. Hyperbolic polar coordinates in the unit disk

We begin by recalling the unit disk as a model of the hyperbolic plane. Thehyperbolic metric on the unit disk D = z : |z| < 1 is

λD(z)|dz| =2|dz|

1 − |z|2 .

The hyperbolic metric has curvature −1; that is,

− log λD(z)

λ2

Ω(z)

= −1,

where z = x + iy and

∆ =∂

2

∂x2+

∂2

∂y2= 4

∂2

∂z∂z


denotes the usual Laplacian. For any piecewise smooth curve γ in D the hyper-bolic length of γ is given by

ℓD(γ) =

∫

γ

λD(z)|dz|.

The hyperbolic distance between z, w ∈ D is defined by

dD(z, w) = inf ℓD(γ),

where the infimum is taken over all piecewise smooth paths γ in D that join z

and w. In fact,

dD(a, b) = 2 tanh−1

∣∣∣∣a − b

1 − ba

∣∣∣∣ .

The group A(D) of conformal automorphisms of the unit disk is the set of holo-morphic isometries of the hyperbolic metric and also of the hyperbolic distance.A path γ joining z to w is called a hyperbolic geodesic arc if dD(z, w) = ℓD(γ).The (hyperbolic) geodesic through z and w is C ∩ D, where C is the unique Eu-clidean circle (or straight line) that passes through z and w and is orthogonal tothe unit circle ∂D. If γ is any piecewise smooth curve joining z to w in D, thenthe hyperbolic length of γ is dD(z, w) if and only if γ is the arc of C in D thatjoins z and w. A hyperbolic disk in the unit disk is DD(a, r) = z : dD(a, z) < r,where a ∈ D is the hyperbolic center and r > 0 is the hyperbolic radius. A hy-perbolic disk in D is Euclidean disk with closure contained in D. In fact, DD(a, r)is the Euclidean disk with center c and radius R, where

c =a

(1 − tanh2(r/2)

)

1 − |a|2 tanh2(r/2)and R =

(1 − |a|2) tanh(r/2)

1 − |a|2 tanh2(r/2).

For more details about hyperbolic geometry on the unit disk the reader shouldconsult [1].

Hyperbolic polar coordinates on the unit disk relative to a specified pole orcenter are defined as follows. Fix a point a in D, called the pole or center forpolar coordinates based at a. For θ in R let ρθ(a, D) = ρθ(a) denote the uniquehyperbolic geodesic ray emanating from a that is tangent to the Euclidean unitvector e

iθ at a. For θ = 0 the Euclidean unit tangent vector is 1 and ρ0(a)is called the horizontal hyperbolic geodesic emanating from a because the unittangent vector at a is horizontal. Of course, ρθ+2nπ(a) = ρθ(a) for all n in Z.Let s 7→ za(s, θ), 0 ≤ s < +∞, be the hyperbolic arc length parametrization ofρθ(a). This means

(2.1)∂za(s, θ)

∂s=

eiΘ(s,θ)

λD(za(s, θ)),

where eiΘ(s,θ) is a Euclidean unit tangent to ρθ(a) at the point za(s, θ). For fixed θ

the point za(s, θ) moves along the geodesic ray ρθ(a) with unit hyperbolic speed.Two hyperbolic geodesic rays with distinct unit tangent vectors at a are disjointexcept for their common initial point and D = ∪ρθ : 0 ≤ θ < 2π. For each z inD \ a there is a unique geodesic ray ρθ(a) with 0 ≤ θ < 2π that contains z, so


there exist unique s > 0 and θ in [0, 2π) with za(s, θ) = z. The hyperbolic polar

coordinates of the point z relative to the center or pole at a are the ordered pair(s, θ), where za(s, θ) = z. The first coordinate, s = dD(a, z), is the hyperbolicdistance from a to z and the second polar coordinate, θ, is the angle between thehorizontal hyperbolic geodesic ray ρ0(a) and the ray ρθ(a) that contains z at thepole a. The hyperbolic circle with hyperbolic center a and hyperbolic radius s iscD(a, s) = z : dD(a, z) = s. Note that each geodesic ray ρθ(a) is orthogonal toevery hyperbolic circle cΩ(a, s). Thus, the coordinate grid for hyperbolic polarcoordinates based at a consists of hyperbolic geodesics emanating from a andhyperbolic circles centered at a. In terms of hyperbolic polar coordinates

λ2

D(z)(dx

2 + dy2) = ds

2 + sinh2(s)dθ2.

For a = 0, ρθ(0) is the radial segment [0, eiθ) with hyperbolic arc length parametriza-tion z0(s, θ) = tanh(s/2)eiθ and

(2.2)∂z0(s, θ)

∂s=

eiθ

λD(z0(s, θ))=

1 − |z0(s, θ)|22|z0(s, θ)|

z0(s, θ).

Hyperbolic polar coordinates about the origin can be transported to any othercenter in the unit disk by a hyperbolic isometry. Recall that each conformalautomorphism of D is an isometry of the hyperbolic metric and the hyperbolicdistance. For a ∈ D the Mobius transformation f(z) = (z + a)/(1 + az) is aconformal automorphism of D that sends the origin to a and f

′(0) = (1−|a|2) > 0.The fact that f

′(0) > 0 insures that f(ρθ(0)) = ρθ(a) for all θ ∈ R and soza(s, θ) = f(z0(s, θ)) provides an explicit hyperbolic arc length parametrizationof ρa(θ):

za(s, θ) =tanh(s/2)eiθ + a

1 + a tanh(s/2)eiθ.

Trivially, the Euclidean distance from a = 0 to z0(s, θ) is an increasing functionof s for each fixed θ and the Euclidean distance between z0(s, θ1) and z0(s, θ2) isan increasing function of s when e

iθ1 6= eiθ2 . It is plausible that these Euclidean

properties remain valid for any center a ∈ D. Rather than investigating theseassertions for the special case of the unit disk, we wait to consider the analogousquestions in any disk or half-plane. Also, hyperbolic circles centered at the originare Euclidean circles.

3. Hyperbolic polar coordinates in a disk or half-plane

We let ∆ denote any Euclidean disk or half-plane when it is not necessaryto distinguish between the cases; otherwise, we use D for a Euclidean disk andH for a Euclidean half-plane. Given ∆ there is a Mobius transformation f thatmaps ∆ onto the unit disk. Then the hyperbolic metric on ∆ is given by

λ∆(z) = λD(f(z))|f ′(z)|.


This defines the hyperbolic density λ∆ independent of the Mobius map of ∆ ontothe unit disk. If D = z : |z − a| < r, then

λD(z)|dz| =2r|dz|

r2 − |z − a|2 .

If H is any half-plane, then

λH(z)|dz| =|dz|

d(z, ∂H),

where d(z, ∂H) denotes the Euclidean distance from z to the boundary of H. Inparticular, for the upper half-plane H = z : Im (z) > 0,

λH(z)|dz| =|dz|

Im (z).

Because Mobius transformations map circles onto circles, hyperbolic geodesicsin a disk or half-plane are arcs of circles orthogonal to the boundary. Also,hyperbolic disks are Euclidean disks with closure contained in the disk or half-plane. Any Mobius map from ∆ onto D is an isometry from ∆ with the hyperbolicmetric to D with the hyperbolic metric. See [1] for details.

Hyperbolic polar coordinates are defined on ∆ analogous to the definitionfor the unit disk. Fix a point w0 in ∆. For θ in R let ρθ(w0, ∆) denote theunique hyperbolic geodesic ray emanating from w0 that is tangent to e

iθ at w0.ρ0(w0, ∆) is called the horizontal hyperbolic geodesic emanating from w0 since itsunit tangent vector at w0 is horizontal. When w0 and ∆ are fixed, we often writeρθ in place of ρθ(w0, ∆). Of course, ρθ+2nπ = ρθ for all n in Z. Let s 7→ w0(s, θ),0 ≤ s < +∞, be the hyperbolic arc length parametrization of ρθ. This means

(3.1)∂w0(s, θ)

∂s=

eiΘ(s,θ)

λ∆(w0(s, θ)),

where eiΘ(s,θ) is a Euclidean unit tangent to ρθ at the point w0(s, θ). Because

∆ = ∪ρθ : 0 ≤ θ < 2π, for each w in ∆\w0 there is a unique geodesic ray ρθ,0 ≤ θ < 2π, that contains w. Hence, there exist unique s > 0 and θ in [0, 2π) withw0(s, θ) = w. The hyperbolic polar coordinates for the point w relative to thecenter or pole at w0 are (s, θ). The coordinate s = d∆(w0, w) is the hyperbolicdistance from w0 to w and θ is the angle between the horizontal hyperbolicgeodesic ray ρ0 and the ray ρθ at w0. The hyperbolic circle with hyperboliccenter w0 and hyperbolic radius s is c∆(w0, s) = w : d∆(w0, w) = s. Thecoordinate grid for hyperbolic polar coordinates consists of hyperbolic geodesicsemanating from w0 and hyperbolic circles centered at w0. If f : D → ∆ is theMobius transformation with f(0) = w0 and f

′(0) > 0, then w0(s, θ) = f(z0(s, θ)).

As we noted in the preceding section when a point in the unit disk movesaway from the origin along a hyperbolic geodesic, the Euclidean distance fromthe origin increases and points along distinct geodesics separate monotonicallyin the Euclidean sense. In fact these properties hold for any disk or half-planeand for any center of hyperbolic polar coordinates.


Theorem 3.1. Let ∆ be any Euclidean disk or half-plane in C and w0 ∈ ∆.

(a) For each θ ∈ R, |w0(s, θ) − w0| is increasing for s ≥ 0.(b) For e

iθ2 6= eiθ1, |w0(s, θ1) − w0(s, θ2)| is an increasing function of s ≥ 0.

Proof. If f : D → ∆ is a Mobius mapping with f(0) = w0 and f′(0) > 0, then

w0(s, θ) = f(z0(s, θ)). Suppose

f(z) =az + b

cz + d,

where ad − bc = 1. Because ∆ is a Euclidean disk or half-plane, ∞ does notlie in ∆. Consequently, −d/c, the preimage of ∞, cannot lie in D; equivalently,|c| ≤ |d|. Since ad − bc = 1, this implies d 6= 0. Also, w0 = f(0) = b/d.

(a) If D(s) = log |w0(s, θ) − w0| = log |f(z0(s, θ)) − w0|, then by using (2.2)we obtain

D′(s) = Re

f′(z0(s, θ))

f(z0(s, θ)) − w0

∂z0(s, θ)

∂s

=1 − |z0(s, θ)|2

2|z0(s, θ)|Re

z0(s, θ)f′(z0(s, θ))

f(z0(s, θ)) − w0

.(3.2)

From

f′(z) =

1

(cz + d)2and f(z) − w0 =

z

d(cz + d),

we obtainzf

′(z)

f(z) − w0

=d

cz + d.

Then for z ∈ D

(3.3) Rezf

′(z)

f(z) − w0

= Redcz + |d|2|cz + d|2 > 0

because |c| ≤ |d| and |z| < 1. Thus, (3.3) and (3.2) imply D(s) is increasing fors ≥ 0, so |w0(s, θ) − w0| is increasing for s ≥ 0.

(b) We assume −π/2 ≤ θ1 = −θ < 0 < θ2 = θ ≤ π/2; the general case can bereduced to this situation by performing a rotation. If

E(s) = log |w0(s, θ) − w0(s,−θ)| = log |f(z0(s, θ)) − f(z0(s,−θ))|,then

E′(s) = Re

f′(z0(s, θ))

∂z0(s,θ)

∂s− f

′(z0(s,−θ))∂z0(s,−θ)

∂s

f(z0(s, θ)) − f(z0(s,−θ)).

Because of (2.2) and |z0(s, θ)| = |z0(s,−θ)|, we obtain

(3.4) E′(s) =

1 − |z0(s, θ)|22|z0(s, θ)|

Re

(z0(s, θ)f

′(z0(s, θ)) − z0(s,−θ)f ′(z0(s,−θ))

f(z0(s, θ)) − f(z0(s,−θ))

).

Direct calculation produces

zf′(z) − ζf

′(ζ)

f(z) − f(ζ)=

d2 − c

2ζz

(cz + d)(cζ + d).


Set t = c/d. Then

zf′(z) − ζf

′(ζ)

f(z) − f(ζ)=

1 − t2ζz

(1 + tz)(1 + tζ)

=1

2

(1 − tζ

1 + tζ+

1 − tz

1 + tz

).

Because (1 − w)/(1 + w) has positive real part for w ∈ D and |tζ|, |tz| < 1, weconclude that for all z, ζ ∈ D

(3.5) Re

(zf

′(z) − ζf′(ζ)

f(z) − f(ζ)

)> 0.

Hence, (3.4) and (3.5) imply E′(s) > 0 for s ≥ 0, so that |w0(s, θ) − w0(s,−θ)|

is an increasing function of s ≥ 0.

4. Hyperbolic polar coordinates in simply connectedregions

A region Ω in the complex plane C is hyperbolic if C \Ω contains at least twopoints. The hyperbolic metric on a hyperbolic region Ω is denoted by λΩ(w)|dw|and is normalized to have curvature

− log λΩ(w)

λ2

Ω(w)

= −1.

If f : D → Ω is any holomorphic universal covering projection, then the densityλΩ of the hyperbolic metric is determined from

(4.1) λΩ(f(z))|f ′(z)| =2

1 − |z|2 .

For a, b in Ω the hyperbolic distance between these points is

dΩ(a, b) = inf

∫

δ

λΩ(w)|dw|,

where the infimum is taken over all piecewise smooth paths δ in Ω joining a andb. A path γ connecting a and b is a hyperbolic geodesic arc if

dΩ(a, b) =

∫

γ

λΩ(w)|dw|.

A hyperbolic geodesic always exists, but need not be unique when Ω is multiplyconnected. Given a ∈ Ω and r > 0, DΩ(a, r) = z ∈ Ω : dΩ(a, z) < r is thehyperbolic disk with hyperbolic center a and hyperbolic radius r.

When Ω is simply connected, any conformal mapping f : D → Ω is an isometryfrom the hyperbolic metric on D to the hyperbolic metric on Ω. In this case f

maps hyperbolic geodesics onto hyperbolic geodesics and hyperbolic disks ontohyperbolic disks. If Ω is multiply connected, then a covering f is only a localisometry, not an isometry.


Suppose Ω is a simply connected hyperbolic region, w0 ∈ Ω and f : D → Ωis the unique conformal mapping with f(0) = w0 and f

′(0) > 0. We can relatehyperbolic polar coordinates on Ω with pole at w0 to those on D with pole atthe origin by using f . Tangent vectors for geodesic rays can be expressed interms of this conformal mapping. Because f is an isometry from the hyperbolicmetric on D to the hyperbolic metric on Ω and f

′(0) > 0, w0(s, θ) = f(z0(s, θ))is the hyperbolic arc length parametrization of ρθ(w0, Ω) and the tangent vectorto ρθ(w0, Ω) is

(4.2)∂w0(s, θ)

∂s=

f′(z0(s, θ))e

iθ

λD(z0(s, θ)).

Thus,

∂w0(0, θ)

∂s=

f′(0)eiθ

2,

so that s 7→ w0(s, θ) is parallel to eiθ at w0. By making use of (4.1) we find

(4.3)∂w0(s, θ)

∂s=

f′(z0(s, θ))

|f ′(z0(s, θ))|e

iθ

λΩ(f(z0(s, θ)))=

ei(ϕ(s,θ)+θ)

λΩ(f(z0(s, θ))),

where eiϕ(s,θ) = f ′

(z0(s,θ))

|f ′(z0(s,θ))| . If arg f′(z) denotes the unique branch of the argument

of f′ that vanishes at w0, then ϕ(s, θ) = arg f

′(z0(s, θ)). From (3.1) and (4.3) weobtain

(4.4) eiΘ(s,θ) = e

i(ϕ(s,θ)+θ).

In a similar manner, hyperbolic disks in Ω are the images of hyperbolic disks in D;explicitly, if f : D → Ω is a conformal map with f(0) = w0, then f(DD(0, r)) =DΩ(w0, r).

5. Euclidean convex univalent functions

Several characterizations of Euclidean convex univalent functions are neededfor our investigation of hyperbolic polar coordinates. We recall two classicalcharacterizations of Euclidean convex and starlike univalent functions. First, alocally univalent holomorphic function f defined on D is a conformal map ontoa Euclidean convex region if and only if [2, p 42]

(5.1) 1 + Rezf

′′(z)

f ′(z)≥ 0

for z ∈ D. Second, if f(0) = w0, then a holomorphic function f defined on D

maps D conformally onto a region starlike with respect to w0 if and only if [2, p41]

(5.2) Rezf

′(z)

f(z) − w0

≥ 0

for z ∈ D.


Theorem 5.1. Suppose f is holomorphic and locally univalent on D. f is Eu-

clidean convex univalent on D if and only if

(5.3) Rezf

′(z) − ζf′(ζ)

f(z) − f(ζ)> 0

for all z, ζ in D.

Proof. We present the short proof. Suppose f is Euclidean convex univalent onD. Then ([27] and [30])

(5.4) Re

2zf ′(z)

f(z) − f(ζ)− z + ζ

z − ζ

> 0

for z, ζ in D. If we interchange the roles of z and ζ in (5.4) and then add thetwo inequalities, we obtain

2 Rezf

′(z) − ζf′(ζ)

f(z) − f(ζ)> 0,

which is equivalent to (5.3).

Conversely, suppose (5.3) holds for all z, ζ in D. Since

limζ→z

zf′(z) − ζf

′(ζ)

f(z) − f(ζ)= 1 +

zf′′(z)

f ′(z),

we obtain (5.1). Hence, f is Euclidean convex univalent on D.

Theorem 5.2. If f is a normalized, f(0) = 0 and f′(0) = 1, Euclidean convex

univalent function on D and θ ∈ (0, π/2], then

(5.5)2|z| sin θ

1 + 2|z| cos θ + |z|2 ≤ |f(eiθz) − f(e−iθ

z)| ≤ 2|z| sin θ

1 − |z|2(

1 + |z|1 − |z|

)cos θ

.

The lower bound is best possible for all θ ∈ (0, π/2] and the upper bound is sharp

for θ = π/2.

Proof. We sketch the idea of the proof. Fix θ in (0, π/2] and consider thefunction

g(z) =f(eiθ

z) − f(e−iθz)

eiθ − e−iθ=

f(eiθz) − f(e−iθ

z)

2i sin θ.

Fromzg

′(z)

g(z)=

eiθzf

′(eiθz) − e

−iθzf

′(e−iθz)

f(eiθz) − f(e−iθz),

Theorem 5.1 implies that g is starlike with respect to the origin on D because(5.2) holds with w0 = 0. If

f(z) = z +∞∑

n=2

anzn,

then

g(z) = z +∞∑

n=2

sin nθ

sin θanz

n.


As f is convex univalent, |a2| ≤ 1 [2]. Hence,

|g′′(0)|2

=

∣∣∣∣sin 2θ

sin θ

∣∣∣∣ |a2| ≤ 2 cos θ.

Then [3] gives the inequalities in (5.5).

Corollary 5.3. If f is a normalized, f(0) = 0 and f′(0) = 1, Euclidean convex

univalent function on D, then for ϕ ∈ (0, π/2]

(5.6)2r

1 + r2≤ |f(reiϕ) − f(−re

iϕ)| ≤ 2r

1 − r2.

These bounds are sharp.

Example 5.4. If K(z) = z/(1 − z), then

K(eiθz) − K(e−iθ

z) =e

iθz

1 − eiθz− e

−iθz

1 − e−iθz

=(2i sin θ)z

1 − (2 cos θ)z + z2.

This shows that the lower bound in (5.5) is sharp for K(z) when z = −r, r is in(0, 1), for any θ in (0, π/2]. For θ = π/2 the upper bound in (5.5) is sharp forthe function K when z = ir, r in (0, 1). Also,

K(r) − K(−r) =2r

1 − r2

and

K(ir) − K(−ir) =2ir

1 + r2,

so both bounds in (5.6) are sharp.

6. Euclidean convex regions

We establish various Euclidean properties for hyperbolic polar coordinates inEuclidean convex regions; in fact, these Euclidean properties characterize con-vex regions. Throughout this section we employ the notation of Section 4. Inparticular, f will always denote a conformal map of D onto Ω with f(0) = w0

and f′(0) > 0. We show that for each fixed θ, the point w0(s, θ) moves mono-

tonically away from w0 in the Euclidean sense. We give sharp upper and lowerbounds on |w0(s, θ) − w0| in terms of s and λΩ(w0). Also, in any convex re-gion Ω distinct hyperbolic geodesic rays separate monotonically in the Euclideansense; this means that for e

iθ2 6= eiθ1 , the distance |w0(s, θ1) − w0(s, θ2)| is an

increasing function of s. We give sharp upper and lower bounds on the difference|w0(s, θ1)−w0(s, θ2)|. These (and other) Euclidean properties of hyperbolic polarcoordinates characterize convex regions.

For example, a classical result of Study [29] implies that for every w0 ∈ Ωeach hyperbolic circle cΩ(w0, s) is a Euclidean convex curve when Ω is convex.The result of Study asserts that if f is a Euclidean convex univalent function,


then f(z : |z| < r) is Euclidean convex for 0 < r < 1. Conversely, if everyhyperbolic circle is Euclidean convex, then Ω is an increasing union of Euclideanconvex regions and so is Euclidean convex.

Theorem 6.1. Let Ω be a simply connected hyperbolic region in C.

(a) If Ω is Euclidean convex and w0 ∈ Ω, then for each θ in R, |w0(s, θ)−w0| is

an increasing function of s and

(6.1)1 − e

−s

λΩ(w0)≤ |w0(s, θ) − w0| ≤

es − 1

λΩ(w0).

These bounds are best possible.

(b) Suppose that for every w0 in Ω and for each θ in R, |w0(s, θ) − w0| is an

increasing function of s. Then Ω is Euclidean convex.

The proof of Theorem 6.1 is given in [16].

Example 6.2. For the upper half-plane H, λH(w) = 1/Im(w). Then for w0 = i,w0(s, π/2) = i + i(es − 1), w0(s,−π/2) = i − i(1 − e

−s) and 1/λH(i) = 1, so theupper and lower bounds are best possible.

Theorem 6.3. Suppose Ω is a simply connected hyperbolic region in C.

(a) If Ω is Euclidean convex, w0 ∈ Ω and eiθ2 6= e

iθ1, then |w0(s, θ1) − w0(s, θ2)|is an increasing function of s ≥ 0 and

(6.2)2 sin θ tanh s

1 + cos θ tanh s≤ |w0(s, θ1) − w0(s, θ2)|λΩ(a) ≤ 2es cos θ sin θ sinh s,

where θ = (θ2 − θ1)/2.(b) If for some w0 in Ω and all e

iθ2 6= eiθ1, |w0(s, θ1)−w0(s, θ2)| is an increasing

function of s ≥ 0, then Ω is Euclidean convex.

Proof. We sketch the proof of (a). First, by translating Ω if necessary, we mayassume w0 = 0. Next, by rotating Ω about the origin if needed, we may assume−π/2 ≤ θ1 = −θ < 0 < θ2 = θ ≤ π/2. Then

|w0(s, θ) − w0(s,−θ)| = |f(z(s, θ)) − f(z(s,−θ))|.All of the quantities involved in the theorem are invariant under translation androtation. If

E(s) = log |w0(s, θ) − w0(s,−θ)| = log |f(z(s, θ)) − f(z(s,−θ))|,then

E′(s) = Re

f′(z(s, θ))∂z(s,θ)

∂s− f

′(z(s,−θ))∂z(s,−θ)

∂s

f(z(s, θ)) − f(z(s,−θ)).

Because of (2.2) and |z(s, θ)| = |z(s,−θ)|, we obtain

(6.3) E′(s) =

1 − |z(s, θ)|22|z(s, θ)| Re

(z(s, θ)f ′(z(s, θ)) − z(s,−θ)f ′(z(s,−θ))

f(z(s, θ)) − f(z(s,−θ))

).

Suppose Ω is Euclidean convex. Then f is a Euclidean convex univalent functionand so (5.3) implies E

′(s) > 0. Hence, |w0(s, θ) − w0(s,−θ)| is an increasingfunction of s ≥ 0. Next, we establish (6.2). The function f/f

′(0) is a normalized


Euclidean convex univalent function so (5.5) with r = tanh(s/2) gives the bounds(6.2) for θ2 = θ and θ1 = −θ since f

′(0) = 2/λΩ(a).

Corollary 6.4. Suppose Ω 6= C is a Euclidean convex region and w0 is a point

of Ω. Then |w0(s, θ) − w0(s, θ + π)| is increasing and

(6.4) tanh(s) ≤ |w0(s, θ) − w0(s, θ + π)|λΩ(a)

2≤ sinh(s).

Both bounds are sharp for a half-plane.

Proof. This is the special case of the theorem in which θ2 = θ + π and θ1 = θ.It corresponds to two hyperbolic geodesic rays emanating from w0 in oppositedirections.

The lower bound in (6.2) has a simple geometric consequence. It gives

lims→+∞

|w0(s, θ1) − w0(s, θ2)| ≥2 tan(θ/2)

λΩ(w0).

For any Euclidean convex region this shows that the ‘ends’ of distinct hyperbolicgeodesic rays emanating from w0 cannot be too close. In particular, (6.4) impliesthat the two ends of a single hyperbolic geodesic cannot be closer than 2/λΩ(w0)for any point w0 on the geodesic. This inequality is sharp for the upper half-plane; we only consider the special case in which θ1 = 0 and θ2 = π. Considerw0 = ib, where b > 0. Then λH(w0) = 1/b. For the hyperbolic geodesic γ throughib that meets R in ±b,

lims→+∞

|w0(s, 0) − w0(s, π)| = 2b =2

λH(w0).

7. Spherical geometry

We discuss the geometry of the spherical plane C∞ with the chordal distanceχ, the spherical metric σ(z) |dz| and the induced spherical distance dσ.

The extended complex plane C∞ is sometimes called the Riemann spherebecause stereographic projection transforms C∞ into the unit sphere. Let S bethe unit sphere x ∈ R

3 : ||x|| = 1 in R3, and let n = (0, 0, 1) be the ‘north

pole’. The stereographic projection ϕ of C∞ onto S is defined as follows. Weregard the complex plane C as a subset of R

3 by identifying z = x + iy with thepoint (x, y, 0). For z in C, the line through z = (x, y, 0) and n meets S at n andat a second point ϕ(z). This defines ϕ on C, and we set ϕ(∞) = n. It is easy tosee that if z = x + iy ∈ C, then

ϕ(x + iy) =

(2x

|z|2 + 1,

2y

|z|2 + 1,|z|2 − 1

|z|2 + 1

).

Observe that ϕ(0) = (0, 0,−1), the south pole, and that ϕ(z) = z = (x, y, 0) ifand only if |z| = 1.


The chordal distance χ is obtained by the following procedure. Use ϕ totransfer points in C∞ to S; then measure the Euclidean distance in R

3 betweenthe image points. Thus, χ is defined by

χ(z, w) = ||ϕ(z) − ϕ(w)||;explicitly,

χ(z, w) =2|z − w|√

(1 + |z|2)(1 + |w|2), χ(z,∞) =

2√(1 + |z|2)

.

This interpretation of χ immediately shows that it is a distance function on C∞.Also, the metric space (C∞, χ) is homeomorphic to S with the restriction of theEuclidean metric; so (C∞, χ) is compact and connected.

The spherical metric on C∞ is given by

σ(w)|dw| =2|dw|

1 + |w|2 ;

it has curvature

−∆ log σ(w)

σ2(w)= 1.

The spherical distance on C∞ derived from this metric is

dσ(z, w) = 2 tan−1

∣∣∣∣z − w

1 + wz

∣∣∣∣ ≤ π.

The chordal and spherical metrics are related to each other by the formula

χ(z, w) = 2 sin

(1

2dσ(z, w)

).

From 2θ/π ≤ sin θ ≤ θ when 0 ≤ θ ≤ π/2, we obtain

(2/π)dσ(z, w) ≤ χ(z, w) ≤ dσ(z, w),

so the two distances induce the same topology on C∞. Note that

limw→z

χ(z, w)

|z − w| = σ(z) = limw→z

dσ(z, w)

|z − w| .

We present a complete description of the isometries of the spherical plane.The orientation preserving conformal isometries of the spherical plane form agroup. All of the following groups are identical,(1) the group of conformal isometries of the chordal distance;(2) the group of conformal isometries of the spherical distance;(3) the group of conformal isometries of the spherical metric;(4) the group of Mobius maps of the form

z 7→ az − c

cz + a, |a|2 + |c|2 = 1.

The orientation-preserving isometries of R3 that fix the origin are the rotations

of R3, and these are represented by the group SO(3) of 3×3 orthogonal matrices

with determinant one. The group SO(3) is conjugate to the group ϕ−1SO(3)ϕ


which acts on C∞; the four identical groups above are equal to ϕ−1SO(3)ϕ. For

this reason the isometries of the spherical plane are sometimes called rotations.

For antipodal z, w ∈ C∞, that is, w = −1/z, any of the infinitely many greatcircular arcs connecting z and w is a spherical geodesic. If z, w ∈ C∞ are notantipodal, then the unique spherical geodesic arc is the shorter arc between z

and w of the unique great circle through z and w.

Just as one studies convex regions in the Euclidean plane it is natural to studyconvex regions in the spherical plane. A simply connected region Ω on C∞ iscalled spherically convex (relative to spherical geometry on C∞) if for each pairof z, w ∈ Ω every spherical geodesic connecting z and w also lies in Ω. If Ω isspherically convex and contains a pair of antipodal points, then Ω = C∞. Ameromorphic and univalent function f defined on D is called spherically convex

if its image f(D) is a spherically convex subset of C∞. A number of authors havestudied spherically convex functions; for example, [6], [8], [11], [15], [19], [21] and[25].

8. Spherically convex univalent functions

In our discussion of Euclidean properties of hyperbolic geodesics, characteri-zations of Euclidean convex functions played a crucial role. Therefore, it is notsurprising that characterizations of spherically convex functions play an impor-tant role in investigating spherical properties of hyperbolic geodesics. One suchcharacterization obtained by Mejia and Minda [19] is

(8.1) Re

1 +

zf′′(z)

f ′(z)− 2zf ′(z)f(z)

1 + |f(z)|2

≥ 0

for all z in D; also see [8]. Sometimes it is difficult to use (8.1) because it con-

tains the nonholomorphic term 2zf ′(z)f(z)/(1 + |f(z)|2). One way to overcomethis difficulty is to establish two-variable characterizations for spherically convexfunctions which are holomorphic in one of the two variables and are analogousto Theorem 5.1

We now state two-variable characterizations for spherically convex functionsthat will be applied to investigate properties of hyperbolic polar coordinateson spherically convex regions and to derive other results for spherically convexfunctions.

Theorem 8.1. Let f be meromorphic and locally univalent in D. Then f is

spherically convex if and only if

(8.2) Re

2zf ′(z)

f(z) − f(ζ)− z + ζ

z − ζ− 2zf ′(z)f(ζ)

1 + f(ζ)f(z)

> 0

for all z, ζ in D.


Proof. Here we prove only the sufficiency. Observe that (8.2) is the sphericalanalog of (5.4). Let

(8.3) p(z, ζ) =2zf ′(z)

f(z) − f(ζ)− z + ζ

z − ζ− 2zf ′(z)f(ζ)

1 + f(ζ)f(z).

We show that if f satisfies the inequality (8.2), then (8.1) holds for all z ∈ D,which characterizes spherically convex functions [19]. The assumption is thatRe p(z, ζ) > 0 for z, ζ ∈ D. Since

(8.4) p(z, z) = 1 +zf

′′(z)

f ′(z)− 2zf ′(z)f(z)

1 + |f(z)|2 ,

f is spherically convex.

Corollary 8.2. Suppose f is meromorphic and locally univalent in D. Then f

is spherically convex if and only if

(8.5) Re

zf

′(z) − ζf′(ζ)

f(z) − f(ζ)− zf

′(z)f(ζ) + ζf ′(ζ)f(z)

1 + f(ζ)f(z)

> 0

for all z, ζ in D.

Proof. Note that (8.5) follows from (8.2) in the same manner that (5.3) wasderived from (5.4). Conversely, suppose (8.5) holds for all z, ζ in D. Set

(8.6) q(z, ζ) =zf

′(z) − ζf′(ζ)

f(z) − f(ζ)− zf

′(z)f(ζ) + ζf ′(ζ)f(z)

1 + f(ζ)f(z).

Then Re q(z, z) > 0 for all z in D. As

q(z, z) = 1 +zf

′′(z)

f ′(z)− 2zf ′(z)f(z)

1 + |f(z)|2 ,

the inequality (8.1) holds. Therefore, f is spherically convex.

As we pointed out earlier, we cannot easily derive properties of sphericallyconvex functions from (8.1) since it contains a nonholomorphic term. WithTheorem 8.1, this difficulty is overcome in some cases. If f is spherically convex,then p(z, ζ) is holomorphic for in z ∈ D, has positive real part, and so satisfies

(8.7)

∣∣∣∣p(z, ζ) − 1 + |z|21 − |z|2

∣∣∣∣ ≤2|z|

1 − |z|2 .

The nonholomorphic function p(z, z) (see (8.4)) still satisfies the inequality (8.7),which holds for the well known class consisting of holomorphic functions p(z) inD with p(0) = 1 and Re p(z) > 0. Note that

∣∣∣∣p(z, z) − 1 + |z|21 − |z|2

∣∣∣∣ ≤2|z|

1 − |z|2also characterizes spherically convex functions and implies the inequality (8.1),see [11].


This idea can be used to derive a number of results for spherically convexfunctions. First, recall that a holomorphic and univalent function f in D withf(0) = f

′(0) − 1 = 0 is called starlike of order β ≥ 0 if Re zf ′(z)/f(z) > β

in D. Using Theorem 8.1, we show that spherically convex functions are closelyrelated to starlike functions.

Theorem 8.3. If f(z) is spherically convex with f(0) = 0, then for every ζ ∈ D,

Fζ(z) =zζ

f(ζ)

f(z) − f(ζ)

(z − ζ)(1 + f(ζ)f(z)

)

is starlike of order 1/2.

Proof. Direct calculations yield

2zFζ(z)

Fζ(z)− 1 = p(z, ζ).

Theorem 8.1 implies the result.

Since Re F (z)/z > 1/2 and F (z)2/z is starlike if F is starlike of order 1/2

(see [26, p. 49]), we get the following results as corollaries of Theorem 8.3.

Corollary 8.4. If f(z) is spherically convex with f(0) = 0, then for every ζ ∈ D,

Re

ζ

f(ζ)

f(z) − f(ζ)

(z − ζ)(1 + f(ζ)f(z)

)

>

1

2

for all z in D.

Corollary 8.5. If f(z) is spherically convex with f(0) = 0, then for every ζ ∈ D,

F2

ζ (z)

z=

zζ2

f(ζ)2

(f(z) − f(ζ))2

(z − ζ)2

(1 + f(ζ)f(z)

)2

is starlike in D.

Mejia and Pommerenke [21] obtained a number of results for spherically con-vex functions by observing that f(z) is (Euclidean) convex when f(z) is spher-ically convex and f(0) = 0. We now provide the sharp order of Euclideanconvexity for spherically convex functions that fix the origin.

Corollary 8.6. Let f(z) = αz + a2z2 + . . ., 0 < α < 1, be spherically convex.

Then for all z in D

Re

1 +

zf′′(z)

f ′(z)

>

(α

1 +√

1 − α2

)2

.

This result is best possible for each α.


Example 8.7. For 0 < α ≤ 1, the spherical half-plane, or hemisphere, Ωα =w : |w −

√1 − α2/α| < 1/α

is spherically convex and

kα(z) =αz

1 −√

1 − α2z

maps D conformally onto Ωα. For the function kα,

inf

Re

1 +

zk′′α(z)

k′α(z)

: z ∈ D

=

(α

1 +√

1 − α2

)2

.

Next, we give the sharp lower bound on Re a2f(z) for normalized sphericallyconvex functions f(z) = αz+a2z

2+. . . . Similar results hold for Euclidean convexfunctions [4] and hyperbolically convex functions [14].

Theorem 8.8. Let f(z) = αz + a2z2 + . . . , 0 < α ≤ 1, be spherically convex.

Then for all z in D

Re a2f(z) ≥ 1 − α2 −

√1 − α2.

This result is best possible for all α.

It is easy to see that for the spherically convex functions kα(z), the infimumof Re a2kα(z) over z ∈ D is 1− α

2 −√

1 − α2, so the lower bound is sharp forall α ∈ (0, 1].

9. Spherically convex regions

Now, we establish certain properties of hyperbolic polar coordinates in spher-ically convex regions. It is convenient to use the density of the hyperbolic metricrelative to the spherical metric; that is,

µΩ(w) =λΩ(w)|dw|σ(w)|dw| =

1

2(1 + |w|2)λΩ(w).

Then λΩ(w)|dw| = µΩ(z)σ(w)|dw| and µΩ is invariant under all rotations of thesphere.

Theorem 9.1. Suppose Ω is a hyperbolic region in C∞.

(a) If Ω is spherically convex and w0 ∈ Ω, then dσ(w0(s, θ), w0) is an increasing

function of s for all θ in R. Moreover, the sharp bounds

tanh(s/2)

µΩ(w0) + tanh(s/2)√

µ2

Ω(w0) − 1

≤ tan1

2dσ(w(s, θ), w0)

≤ tanh(s/2)

µΩ(w0) − tanh(s/2)√

µ2

Ω(w0) − 1

.

hold.

(b) If dσ(w0(s, θ), w0) is an increasing function of s for each w0 in Ω and all θ

in R, then Ω is spherically convex.

The proof of Theorem 9.1 is given in [17].


Example 9.2. Consider the function kα defined in Example 8.7. For w0 = 0,µΩα

(w0) = 1/α,

w0(s, 0) =α tanh(s/2)

1 − tanh(s/2)√

1 − α2

is the hyperbolic arc length parametrization of [0, 1+√

1−α2

α), and the upper bound

is equal toα tanh(s/2)

1 − tanh(s/2)√

1 − α2= tan

1

2dσ(w0(s, 0), 0).

This shows that the upper bound is sharp. Similarly,

w0(s, π) =−α tanh(s/2)

1 + tanh(s/2)√

1 − α2

is the hyperbolic arc length parametrization of (−1+√

1−α2

α, 0], and the lower bound

is equal toα tanh(s/2)

1 + tanh(s/2)√

1 − α2= tan

1

2dσ(w0(s, π), 0).

Hence, the lower bound is also sharp.

Theorem 9.3. Suppose Ω is a hyperbolic region in C∞.

(a) If Ω is spherically convex and w0 ∈ Ω, then dσ(w0(s, θ1), w0(s, θ2)) is an in-

creasing function of s whenever eiθ2 6= e

iθ1.

(b) If there exists w0 ∈ Ω such that dσ(w0(s, θ1), w0(s, θ2)) is an increasing func-

tion of s whenever eiθ2 6= e

iθ1, then Ω is spherically convex.

The reader can consult [17] for a proof of Thorem 9.3.

Geometrically, Theorem 9.3(a) indicates that in a spherically convex regionΩ, two hyperbolic geodesics starting off in different directions from a point w0 inΩ will spread farther apart relative to the spherical distance.

If Ω is spherically convex, then so is every hyperbolic disk and conversely.This follows from the analog of Study’s theorem for spherically convex functions;see [19].

10. Hyperbolic geometry

In this section, we indicate similar monotonicity properties for hyperbolicpolar coordinates in hyperbolically convex regions. Because of the numeroussimilarities with the Euclidean and spherical cases, we present even fewer detailsin this situation. It is convenient to introduce the notation

νΩ(w) =λΩ(w)|dw|λD(w)|dw| =

1

2(1 − |w|2)λΩ(w)

for the density of the hyperbolic metric of a region Ω ⊂ D relative to the back-ground hyperbolic metric λD(w)|dw|.

A simply connected region Ω in D is called hyperbolically convex (relative tothe background hyperbolic geometry on D) if for all points z, w ∈ Ω the arc of


the hyperbolic geodesic in D connecting z and w also lies in Ω. A holomorphicand univalent function f defined on D with f(D) ⊂ D is called hyperbolically

convex if its image f(D) is a hyperbolically convex subset of D. Hyperbolicallyconvex functions have been studied by a number of authors [7], [8], [12], [13],[14], [17], [20], [22]. The related concept of hyperbolically 1-convex functionswas investigated in [9].

There are known characterizations of hyperbolically convex functions. Forexample, a holomorphic and locally univalent function f with f(D) ⊂ D is hy-perbolically convex if and only if [12]

(10.1) Re

1 +

zf′′(z)

f ′(z)+

2zf ′(z)f(z)

1 − |f(z)|2

≥ 0

for all z in D. Mejia and Pommerenke [22] (also see [14]) showed that a holomor-phic and locally univalent function f with f(D) ⊂ D is hyperbolically convex ifand only if

(10.2) Re

2zf ′(z)

f(z) − f(ζ)− z + ζ

z − ζ+

2zf ′(z)f(ζ)

1 − f(ζ)f(z)

> 0

for all z, ζ in D. This is the hyperbolic analog of (8.2). Similar to the proof ofCorollary 8.2, we obtain the following characterization from (10.2).

Theorem 10.1. A holomorphic and locally univalent function f with f(D) ⊂ D

is hyperbolically convex if and only if

(10.3) Re

zf

′(z) − ζf′(ζ)

f(z) − f(ζ)+

zf′(z)f(ζ) + ζf ′(ζ)f(z)

1 − f(ζ)f(z)

> 0

for all z, ζ in D.

These two-point characterizations can be used to derive monotonicity proper-ties of hyperbolic polar coordinates on hyperbolically convex regions in D.

Theorem 10.2. Let Ω ⊂ D.

(a) If Ω is hyperbolically convex and w0 ∈ Ω, then dD(w0(s, θ), w0) is an increasing

function of s for all θ in R. Moreover, we have the following sharp bounds:

2 tanh(s/2)

νΩ(w0)(1 + tanh(s/2)) +√

ν2

Ω(w0)(1 + tanh(s/2))2 − 4 tanh(s/2)

≤ tanh1

2dD(w0(s, θ), w0)

≤ 2 tanh(s/2)

νΩ(w0)(1 − tanh(s/2)) +√

ν2

Ω(w0)(1 − tanh(s/2))2 + 4 tanh(s/2)

.

(b) If dD(w0(s, θ), w0) is an increasing function of s for each w0 in Ω and all θ

in R, then Ω is hyperbolically convex.


Example 10.3. For 0 < α ≤ 1, the hyperbolic half-plane

Hα = D \

w :

∣∣∣∣w +1

α

∣∣∣∣ ≤√

1 − α2

α

is hyperbolically convex and

Kα(z) =2αz

1 − z +√

(1 − z)2 + 4α2z

maps D conformally onto Hα. When w0 = 0, νHα(0) = 1/α, w0(s, 0) = Kα(tanh(s/2))

is the hyperbolic arc length parametrization of [0, 1), and the upper bound isequal to

2α tanh(s/2)

1 − tanh(s/2) +√

(1 − tanh(s/2))2 + 4α2 tanh(s/2)= tanh

1

2dD(w0(s, 0), 0).

This shows that the upper bound is sharp. Similarly, w0(s, π) = kα(− tanh(s/2))

is the hyperbolic arc length parametrization of (−1−√

1−α2

α, 0], and the lower bound

is equal to

2α tanh(s/2)

1 + tanh(s/2) +√

(1 + tanh(s/2))2 − 4α2 tanh(s/2)= tanh

1

2dD(w0(s, π), 0).

Thus, the lower bound is also sharp.

The proof of Theorem 10.4 below is analogous to the proof of Theorem 9.3;the characterization (10.3) for hyperbolically convex functions is used in place of(8.2).

Theorem 10.4. Suppose Ω ⊂ D.

(a) If Ω is hyperbolically convex and w0 ∈ Ω, then dD(w0(s, θ1), w0(s, θ2)) is an

increasing function of s for all eiθ2 6= e

iθ1.

(b) If there exists w0 ∈ Ω such that dD(w0(s, θ1), w0(s, θ2)) is an increasing func-

tion of s whenever eiθ2 6= e

iθ1, then Ω is hyperbolically convex.

Ω ⊂ D is hyperbolically convex if and only if every hyperbolic disk DΩ(w0, r)is hyperbolically convex as a subset of D. This is a direct consequence of theanalog of Study’s Theorem for hyperbolically convex functions; see [20] and [12].

11. Concluding remarks

Relative to the background geometry (hyperbolic, Euclidean, or spherical)hyperbolic geodesics and hyperbolic disks have similar behavior in convex re-gions. Moreover, there are numerous similarities between conformal maps of theunit disk onto convex regions in each of the three geometries, although formu-las for spherical or hyperbolic convexity can be more complicated than thosefor Euclidean convexity because the spherical plane and the hyperbolic planehave nonzero curvature. It is possible to obtain results for Euclidean convex-ity as a limit of corresponding results for spherical convexity; for example, see


Kim-Minda [6] for an illustration of the method. In the same manner Euclideanresults can be obtained as the limit of hyperbolic convexity results.

Holomorphic functions defined on the unit disk can be viewed as maps from D

to the Euclidean plane. Bounded holomorphic functions on the unit disk can beregarded as maps from D to the hyperbolic plane, provided they are scaled to bebounded by one. Finally, meromorphic functions on D can be considered as mapsinto the spherical plane. Sometimes connections between classical results can bemade by adopting this geometric view. This paper showed the close connectionbetween Euclidean convexity, spherical convexity and hyperbolic convexity. Also,by adopting this geometric viewpoint it is possible to recognize there should beanalogs of classical results for holomorphic functions for maps into the other twogeometries.

Some other function theory papers that relate to comparisons between hyper-bolic, Euclidean and spherical geometry are [5], [23], [24], [25].

References

[1] A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Pro-ceedings of the International Workshop on Quasiconformal Mappings and their Applica-tions, Narosa Publishing House, India, (2006), 159-206.

[2] P. Duren, Univalent functions, Grundlehern Math. Wiss. 259, Springer, New York 1983.[3] M. Finkelstein, Growth estimates for convex functions, Proc. Amer. Math. Soc. 18 (1967),

412-418.[4] R. Fournier, J. Ma and S. Ruscheweyh, Convex univalent functions and omitted values,

Approximation Theory, 21 (1998), 225-241.[5] S. Kim and D. Minda, A geometric approach to two-point comparisons for hyperbolic and

euclidean geometry on convex regions, J. Korean Math. Soc., 36 (1999), 1169-1180.[6] S. Kim and D. Minda, The hyperbolic metric and spherically-convex regions, J. Math.

Kyoto Univ., 41 (2001), 285-302.[7] S. Kim and T. Sugawa, Characterizations of hyperbolically convex regions, J. Math. Anal.

Appl. 309 (2005), 37-51.[8] W. Ma, D. Mejia and D. Minda, Distortion theorems for hyperbolically and spherically

k-convex functions, Proc. of an International Conference on New Trends in GeometricFunction Theory and Application, R. Parvathan and S. Ponnusamy (editors), World Sci-entific, Singapore, 1991, 46-54.

[9] W. Ma, D. Mejia and D. Minda, Hyperbolically 1-convex functions, Ann. Polon. Math.,84 (2004), 185-202.

[10] W. Ma and D. Minda, Euclidean linear invariance and uniform local convexity, J. Austral.Math. Soc. 52 (1992), 401-418.

[11] W. Ma and D. Minda, Spherical linear invariance and uniform local spherical convexity,Current Topics in Geometric Function Theory, H. M. Srivastava and S. Owa (editors),World Scientific, Singapore, 1993, 148-170.

[12] W. Ma and D. Minda, Hyperbolically convex functions, Ann. Polonici Math. 60 (1994),81-100.

[13] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Aus-tralian Math. Soc., 58 (1995), 73-93.

[14] W. Ma and D. Minda, Hyperbolically convex functions II, Ann. Polonici Math. 71 (1999),273-285.

[15] W. Ma and D. Minda, Two-point distortion theorems for spherically-convex functions,Rocky Mtn. J. Math., 30 (2000), 663-687.


[16] W. Ma and D. Minda, Euclidean properties of hyperbolic polar coordinates, Comput.Methods Funct. Theory, to appear.

[17] W. Ma and D. Minda, Geometric properties of hyperbolic polar coordinates, submitted.[18] D. Mejia and D. Minda, Hyperbolic geometry in k-convex regions, Pacific J. Math., 141

(1990), 333-354.[19] D. Mejia and D. Minda, Hyperbolic geometry in spherically k-convex regions, Compu-

tational Methods and Function Theory, Proceedings, Valparaiso, Chili, S. Ruscheweyh,E. B. Saff, L. C. Salinas and R. S. Varaga (editors), Lecture Notes in Mathematics, Vol.1435, Springer-Verlag, New York, 1990, 117-129.

[20] D. Mejia and D. Minda, Hyperbolic geometry in hyperbolically k-convex regions, Rev.Columbiana Math., 25 (1991), 123-142.

[21] D. Mejia and Ch. Pommerenke, On spherically convex functions, Michigan Math. J. 47

(2000), 163-172.[22] D. Mejia and Ch. Pommerenke, On hyperbolically convex functions, J. Geom. Anal. 10

(2000), 365-378.[23] D. Minda, Bloch constants, J. Analyse Math. 4 (1982), 54-84.[24] D. Minda, Estimates for the hyperbolic metric, Kodai Math. J., 8 (1985), 249-258.[25] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J.

Analyse Math., 49 (1987), 90-105.[26] R. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l’Universite

de Montreal, Montreal, 1982.[27] T. Sheil-Small, On convex univalent functions, J. London Math. Soc. 1 (1969), 483-492.[28] D. J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, Cambridge

1950.[29] E. Study, Konforme Abbildung einfachzusammenhangerder Bereiche, Vorlesungen uber

ausgewahlte Gegenstande, Heft 2, Leipzig und Berlin 1913.[30] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970),

775-777.

W. Ma E-mail: [email protected]: School of Integrated Studies, Pennsylvania College of Technology, Williamsport, PA

17701, USA

D. Minda E-mail: [email protected]: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio

45221-0025, USA


Quasiminimizers and Potential Theory

Olli Martio

Abstract. Quasiminimizers are almost minimizers of variational integrals.Although quasiminimizers do not form a sheaf and do not provide a uniquesolution to the Dirichlet problem it is shown that they form an interesting basisfor a potential theory. Quasisuperminimizers and their Poisson modificationsare considered as well as their convergence properties. Special attention isdevoted to the theory on the real line.

Keywords. quasiminimizers, quasisuperminimizers, quasisuperharmonic func-tions.

2000 MSC. Primary: 31C45; Secondary: 35J20, 35J60.

Contents

1. Introduction 189

2. Case n = 1 191

3. Properties of quasiminimizers 193

4. Quasisuperminimizers, Poisson modifications and regularity 194

5. More about n = 1 198

6. Quasisuperharmonic functions 199

7. Appendix 1 200

8. Appendix 2 202

References 205

1. Introduction

Quasiminimizers minimize a variational integral only up to a multiplicativeconstant. More precisely, let Ω ⊂ R

n be an open set, K ≥ 1 and 1 ≤ p < ∞. Inthe case of the p-Dirichlet integral, a function u belonging to the Sobolev spaceW

1,ploc

(Ω) is a (p,K)-quasiminimizer or a K-quasiminimizer, if

(1.1)

∫

Ω′

|∇u|pdx ≤ K

∫

Ω′

|∇v|p dx

Version October 19, 2006.

190 O. Martio IWQCMA05

for all functions v ∈ W1,p(Ω′) with v − u ∈ W

1,p0

(Ω′) and for all open sets Ω′

with a compact closure in Ω. A 1-quasiminimizer, called a minimizer, is a weaksolution of the corresponding Euler equation

(1.2) div(|∇u|p−2∇u) = 0.

Clearly being a weak solution of (1.2) is a local property. However, being a K-quasiminimizer is not a local property as one-dimensional examples easily show.This indicates that the theory for quasiminimizers differs from the theory forminimizers and that there are some unexpected difficulties.

Quasiminimizers have been previously used as tools in studying the regular-ity of minimizers of variational integrals, see [GG1–2]. The advantage of thisapproach is that it covers a wide range of applications and that it is based onlyon the minimization of the variational integrals instead of the corresponding Eu-ler equation. Hence regularity properties as Holder continuity and L

p-estimatesare consequences of the quasiminimizing property. It is an important fact thatnonnegative quasiminimizers satisfy the Harnack inequality, see [DT].

Instead of using quasiminimizers as tools, the objective of these lectures is toshow that quasiminimizers have a fascinating theory themselves. In particular,they form a basis for nonlinear potential theoretic model with interesting fea-tures. From the potential theoretic point of view quasiminimizers have severaldrawbacks: They do not provide unique solutions of the Dirichlet problem, theydo not obey the comparison principle, they do not form a sheaf and they donot have a linear structure even when the corresponding Euler equation is lin-ear. However, quasiminimizers form a wide and flexible class of functions in thecalculus of variations under very general circumstances. Observe that the quasi-minimizing condition (1.1) applies not only to one particular variational integralbut the whole class of variational integrals at the same time. For example, if avariational kernel F (x,∇u) satisfies

(1.3) α|h|p ≤ F (x, h) ≤ β|h|p

for some 0 < α ≤ β < ∞, then the minimizers of∫

F (x,∇u) dx

are quasiminimizers of the p-Dirichlet integral

(1.4)

∫|∇u|p dx.

Hence the potential theory for quasiminimizers includes all minimizers of allvariational integrals similar to (1.4). The essential feature of the theory is thecontrol provided by the bounds in (1.3).

For example, the coordinate functions of a quasiconformal or, more generally,quasiregular mapping are quasiminimizers of the n-Dirichlet integral

∫|∇u|n dx

Quasiminimizers and Potential Theory 191

in all dimensions n = 2, 3, . . . .

Recently quasiminimizers have been considered in metric measure spaces. Thismeans that a metric space (X, d) is equipped with a Borel measure µ whichsatisfies some standard assumptions like the doubling property. The Sobolevspace W

1,p is replaced by the so called Newtonian space N1,p which for R

n andthe Lebesgue measure reduces to W

1,p. We do not consider metric spaces herealthough most of the results hold in this case under appropriate conditions. Forthis theory see [KM2].

2. Case n = 1

For n = 1 the definition (1.1) can be written in the following form: Let (a, b)be an open interval in R and u ∈ W

1,ploc

(a, b). Then u is a (p,K)-quasiminimizer,or K-quasiminimizer for short, if for all closed intervals [c, d] ⊂ (a, b)

(2.1)

d∫

c

|u′|p dx ≤ K

d∫

c

|v′|p dx

whenever u − v ∈ W1,p0

(c, d).

Now affine functions are minimizers, i.e. 1-quasiminimizers, for every p ≥ 1.This fact can be easily deduced from the one dimensional version of (1.2) if p > 1.For p = 1 this is trivial. Moreover, affine functions are the only minimizers forp > 1. Thus choosing v(x) = α(x − c) + β where

α = (u(d) − u(c))/(d − c), β = u(c)

we see that u − v ∈ W1,p0

(c, d) and (2.1) yields

(2.2)

d∫

c

|u′|pdx ≤ K|u(d) − u(c)|p|d − c|p−1

,

see [GG2]. The inequality (2.2) gives another definition for a K-quasiminimizeru: the function u is a locally absolutely continuous function in (a, b) that satisfies(2.2) on each subinterval [c, d] of (a, b).

Observe that u ∈ W1,p(c, d) in a bounded open interval (c, d) means that u is

absolutely continuous on [c, d] with

(2.3)

d∫

c

|u′|pdx < ∞.

If u ∈ W1,p(c, d) and u − v ∈ W

1,p0

(c, d), then v ∈ W1,p(c, d) and v(c) =

u(c), v(d) = u(d). Functions u ∈ W1,ploc

(a, b) are simply locally absolutely contin-uous functions on (a, b) such that (2.3) holds in each subinterval [c, d] ⊂ (a, b).

We leave the following lemma as an exercise.


Lemma 2.4. Suppose that u is a (p,K)-quasiminimizer in (a, b). Then u is a

monotone function. If p > 1 then u is either strictly monotone or constant.

The following lemma is more difficult to prove. It does not hold for p = 1.

Lemma 2.5. Let u be a (p,K)-quasiminimizer, p > 1, in an interval (a, b). If

b < ∞, then u has a continuous extension to b and (2.2) holds in all intervals

[c, d] ⊂ (a, b].

Proof. We may assume that u is increasing, b = 1, [0, 1] ⊂ (a, b] and u(0) = 0.

Fix 0 ≤ c < t < 1. Nowt∫

c

u′dx ≤ (t − c)(p−1)/p

( t∫

c

u′p

dx

) 1

p

≤ (t − c)(p−1)/p

( t∫

0

u′p

dx

) 1

p

≤ K1

p

(t − c)p−1

p

tp−1

p

t∫

0

u′dx = K

1

p

(1 − c

t

) p−1

p

t∫

0

u′dx

where we have used the Holder inequality and (2.2). Next we choose c = 1 −(2p

K)1

1−p . Then 0 < c < 1 and letting t ∈ (c, 1) we obtain(

1 − c

t

) p−1

p

< (1 − c)p−1

p =1

2K1

p

.

The above inequalities yieldt∫

0

u′dx =

c∫

0

u′dx +

t∫

c

u′dx ≤

c∫

0

u′dx +

1

2

t∫

0

u′dx

and hence

u(t) = u(t) − u(0) =

t∫

0

u′dx ≤ 2

c∫

0

u′dx = 2u(c).

Since u is increasing, letting t → 1 we obtain

u(b) = u(1) = limt→1

u(t) ≤ 2u(c) < ∞

and the last assertion of the lemma now follows from (2.2).

Lemma 2.5 shows that the natural domain of definition for a 1-dimensionalquasiminimizer is the closed interval [a, b].

Example 2.6. The function u(x) = xα, α > 1/2, is a (2, K)-quasiminimizer

in [0,∞) for K = α2/(2α − 1). This is a rather easy computation. Note that

u(x) = x1/2 is not a (2, K)-quasiminimizer in [0,∞) (and in (0,∞)) since u

′ doesnot belong to L

2(0, 1).

We will consider one dimensional quasiminimizers again in Chapter 5. Theone dimensional case was first studied in [GG2].


3. Properties of quasiminimizers

We start with a basic regularity property.

Theorem 3.1. Suppose that u is a (p,K)-quasiminimizer in Ω ⊂ Rn, p > 1. If

0 < r < R are such that the ball B(x, 2R) ⊂ Ω, then

osc(u,B(x, r)) ≤ C(r/R)αosc(u,B(x,R))

where C < ∞ and α ∈ (0, 1] depend on p, n and K only.

In particular Theorem 3.1 implies that u is locally Holder continuous in Ω.

Another important property is the Harnack inequality.

Theorem 3.2. Let u be as in Theorem 3.1 and u ≥ 0. Then in each ball B(x,R)such that B(x, 2R) ⊂ Ω

supB(x,R)

u ≤ C infB(x,R)

u

where the constant C depends on p, n and K only.

Since quasiminimizers are functions in W1,ploc

(Ω) only, Theorem 3.1 also meansthat they can be made continuous after redefinition on a set of measure zero.

For p > n, and hence for all p > 1 for n = 1, Theorem 3.1 follows fromthe Sobolev embedding lemma. Note that for p = 1 = n quasiminimizers arecontinuous but they need not be Holder continuous.

We do not prove Theorems 3.1 and 3.2 here. The proof for Theorem 3.2 inthe case n = 1 is relatively simple, see [GG2]. For the proof of Theorem 3.1 theDe Giorgi method can be used. The basic tool is the Sobolev type inequality

( ∫−

B(x,r)

|u|t dx

)1/t

≤ cr

( ∫−

B(x,r)

|∇u|p dx

)1/p

for functions u ∈ W1,p0

(B(x, r)) where t > p. The main difficulty is to prove thatu is locally essentially bounded. For the proof see [GG1], [GG2] and [KS]. In thepaper [KS] metric measure spaces are considered and hence the regularity proofof [KS] uses minimal assumptions.

In the general case n ≥ 2 the proof for Theorem 3.2 is rather complicated, see[DT] and [KS]. The proof makes use of the Krylov–Safonov covering argument[KSa]. Very recently it has turned out that the Moser method can be employed toprove Theorems 3.1 and 3.2 for quasiminimizers even in metric measure spaces,see [Ma].

In Potential Theory the Harnack inequality and Harnack’s principle are essen-tially equivalent. From Theorem 3.1 and 3.2 it easily follows (p > 1): Supposethat (ui) is an increasing sequence of K-quasiminimizers in a domain Ω. Iflim ui(x0) < ∞ at some point x0 ∈ Ω, then lim ui is a K-quasiminimizer.

We will return to the proof of this fact in the next chapter and in Appendix 2.


4. Quasisuperminimizers, Poisson modifications andregularity

Let Ω be an open set in Rn. A function u ∈ W

1,ploc

(Ω) is called a (p,K)-quasisuperminimizer, or a K-quasisuperminimizer, if

(4.1)

∫

Ω′

|∇u|p dx ≤ K

∫

Ω′

|∇v|p dx

holds for all open Ω′ ⊂⊂ Ω and all v ∈ W1,ploc

(Ω) such that v ≥ u a.e. in Ω′ and

v − u ∈ W1,p0

(Ω′).

Remarks 4.2. (a) A 1-quasisuperminimizer is called a superminimizer.

(b) A superminimizer is a supersolution of the p-harmonic equation

∇ · (|∇u|p−2∇u) = 0,

i.e. u satisfies ∫

Ω

|∇u|p−2∇u · ∇ϕdx ≥ 0

for all non–negative ϕ ∈ C∞0

(Ω), see [HKM] for this theory. Observe that forp = 2 every superharmonic (in the classical sense) function u is a superminimizerprovided that u belongs to W

1,2loc

(Ω), however, a superharmonic function need not

belong to W1,2loc

(Ω). For n = 2 the classical example is u(x) = − log |x| which issuperharmonic in R

2 but does not belong to W1,2(B(0, 1)). We return to this

problem in Chapter 6.

(c) The inequality (4.1) can be replaced by several other inequalities, forexample ∫

Ω′\E

|∇u|p dx ≤ K

∫

Ω′\E

|∇v|p dx,

where E ⊂ Ω′ \ u 6= v is any measurable set. For the list of these conditionssee [B] and [KM2].

A function u is called a K-quasisubminimizer if −u is a K-quasisuperminimizer.

Note that if u is a K-quasisuperminimimizer, then αu and u + β are K-quasisuperminimizers when α ≥ 0 and β ∈ R. However, the sum of two K-quasisuperminimizers need not be a K-quasisuperminimizer even in the casep = 2.

Lemma 4.3. Suppose that ui, i = 1, 2, are Ki-quasisuperminimizers in Ω. Then

min(u1, u2) is a K-quasisuperminimizer in Ω with K = min(K1K2, K1 + K2).

Proof. We prove that u = min(u1, u2) is a K-quasisuperminimizer with K ≤K1K2; this inequality is important in applications. The proof for K ≤ K1 + K2

is similar, see [KM2]. To this end let Ω′ ⊂⊂ Ω be an open set and v − u ∈W

1,p0

(Ω′), v = u in Ω \ Ω′ and v ≥ u. Now


∫

Ω′

|∇u|p dx =

∫

u1≤u2∩Ω′

|∇u1|p dx +

∫

u1>u2∩Ω′

|∇u2|p dx

and write w = max(min(u2, v), u1). Then w ≥ u1 a.e. in Ω′, w − u1 ∈ W

1,p0

(Ω′)and w = u1 if u1 > u2. Thus the quasisuperharmonicity of u1, see Remark 4.2(c), yields∫

u1≤u2∩Ω′

|∇u1|p dx ≤ K1

∫

u1≤u2∩Ω′

|∇w|p dx

= K1

∫

u1≤u2∩v<u2∩Ω′

|∇v|p dx + K1

∫

u1≤u2∩v≥u2∩Ω′

|∇u2|p dx.

¿From these inequalities we obtain∫

Ω′

|∇u|p dx ≤ K1

∫

u1≤u2∩v<u2∩Ω′

|∇v|p dx

+ K1

∫

u1≤u2∩v≥u2∩Ω′

|∇u2|p dx +

∫

u1>u2∩Ω′

|∇u2|p dx

≤ K1

∫

u1≤u2∩v<u2∩Ω′

|∇v|p dx + K1

∫

v≥u2∩Ω′

|∇u2|p dx.

Note that Ω′ ∩u1 > u2 ⊂ Ω′ ∩v ≥ u2. On the other hand max(u2, v)− u2 ∈W

1,p0

(Ω′), max(u2, v) ≥ u2 and max(u2, v) − u2 = 0 in v ≤ u2 and hence thequasisuperharmonicity of u2 implies

∫

v≥u2∩Ω′

|∇u2|p dx ≤ K2

∫

v≥u2∩Ω′

|∇v|p dx.

This together with the previous inequality completes the proof.

The following corollary is important.

Corollary 4.4. Suppose that u is a K-quasisuperminimizer and h is a super-

minimizer in Ω. Then min(u, h) is a K-quasisuperminimizer in Ω.

Remark 4.5. Lemma 4.3 and Corollary 4.4 are the counterparts of a prop-erty of classical superharmonic functions: If u1 and u2 are superharmonic, thenmin(u1, u2) is superharmonic.

Corollary 4.4 implies the necessity part of the following result. The other halffollows from Theorem 4.14 below.

Lemma 4.6. Suppose that u ∈ W1,ploc

(Ω). Then u is a K-quasisuperminimizer,

p > 1, if and only if min(u, c) is a K-quasisuperminimizer for each c ∈ R.


The Poisson modification is an important tool in Potential Theory. In theclassical case p = 2 this means the following: Let u be superharmonic in Ωand u ∈ W

1,2loc (Ω) (this assumption is not actually needed). If Ω′ ⊂⊂ Ω is an

open set let h be a minimizer (harmonic) in Ω′ with boundary values u, i.e.u − h ∈ W

1,20

(Ω′). The Poisson modification of u in Ω′ is defined as

(4.7) P (u, Ω′) =

h in Ω′

,

u in Ω \ Ω′.

Then P (u, Ω′) is a superharmonic function in Ω, P (u, Ω′) ≤ u and P (u, Ω′) isharmonic in Ω′. This theory works well for superminimizers for all p > 1, see[HKM].

For quasisuperminimizers the above method does not work as above. In partic-ular there exists a K-quasisuperminimizer (even a K-quasiminimizer) u such thatthe function P (u, Ω′) in (4.7) is not a K

′-quasisuperminimizer for any K′< ∞.

The example is one dimensional and requires some computation. However, thereare two replacements for P (u, Ω′).

The first Poisson modification is a modification of (4.7). Let u be a K-quasisuperminimizer in Ω, p > 1, and Ω′ ⊂⊂ Ω an open set. Let h be theminimizer with boundary values u in Ω′, i.e. u − h ∈ W

1,p0

(Ω′). Observe thatsuch a unique function h exists - this is a basic result in the theory of Sobolevspaces, see e.g. [HKM]. Let

(4.8) P1(u, Ω′) =

min(u, h) in Ω′

,

u in Ω \ Ω′.

Theorem 4.9. [KM2] The function P1(u, Ω′) is K-quasisuperminimizer in Ω.

By the construction of P1(u, Ω′), P1(u, Ω′) ≤ u in Ω. Note also that if u is aK-quasiminimizer, then P1(u, Ω′) is also a K-quasiminimizer in Ω′ by Corollary4.4.

Next we consider another possibility for a Poisson modification. For this weneed to consider obstacle problems. The obstacle method is the most importantmethod in the nonlinear potential theory. Let Ω ⊂ R

n be an open set andu ∈ W

1,p(Ω). Write

K+

u (Ω) = v ∈ W1,p(Ω) : v − u ∈ W

1,p0

(Ω), v ≥ u a.e. in Ω.The class K−

u is defined similarly but v ≥ u is replaced by v ≤ u. The followingresult is well-known.

Lemma 4.10. [HKM] The obstacle problem

infv∈K−

u (Ω)

∫

Ω

|∇v|pdx, p > 1,

has a unique solution u− ∈ K−

u (Ω). Moreover, u− is a subminimizer and contin-

uous if u is continuous.


A similar result holds for the class K+

u (Ω) and the solution is a superminimizer.

Suppose now that u is a K-quasisuperminimizer in Ω and Ω′ ⊂⊂ Ω is open.Let u

− be the solution to the K−u (Ω′)-obstacle problem. Define

(4.11) P2(u, Ω′) =

u− in Ω′

,

u in Ω \ Ω′.

Theorem 4.12. The function P2(u, Ω′) is a K-quasisuperminimizer in Ω, a

subminimizer in Ω′ and a K-quasiminimizer in Ω′.

The proof for Theorem 4.12 is in Appendix 1.

Superharmonic functions in the classical potential theory are lower semicon-tinuous. It turns out that quasisuperminimizers can be defined pointwise andthe resulting function is lower semicontinuous.

Theorem 4.13. [KM2] Suppose that u is a K-quasisuperminimizer in Ω, p > 1.Then the function u

∗ : Ω → (−∞,∞] defined by

u∗(x) = lim

r→0

ess infB(x,r)

u

is lower semicontinuous and u∗ = u a.e. (u and u

∗ are the same function in

W1,nloc

(Ω)).

The proof for Theorem 4.13 is based on the De Giorgi method that is used toprove a weak Harnack inequality

( ∫−

B(x,r)

uσdx

)1/σ

≤ c ess infB(x,3r)

u

for a K-quasisuperharmonic function u ≥ 0. Here B(x, 5r) ⊂ Ω and c and σ > 0depend only on n, p > 1 and K.

The following is Harnack’s principle for quasisuperminimizers.

Theorem 4.14. Suppose that (ui) is an increasing sequence of K-quasisupermini-

mizers in Ω, p > 1. If u = lim ui is such that either

(i) u is locally bounded above or

(ii) u ∈ W1,ploc

(Ω),

then u is a K-quasisuperminimizer.

The proof for Theorem 4.14 is somewhat tedious. It is presented in Appen-dix 2.


5. More about n = 1

We take a closer look at the case n = 1. Recall that a superminimizer is a1-quasisuperminimizer. The next result is easy to prove, see [MS].

Lemma 5.1. Suppose that u : [a, b] → R is a superminimizer, p > 1. Then u is

a concave function.

¿From Lemma 5.1 it follows that a superminimizer is a Lipschitz function ineach interval [c, d] ⊂⊂ (a, b). It need not be a Lipschitz function in [a, b].

How regular are K-quasiminimizers and K-quasisuperminimizers? The an-swer is not known for n ≥ 2 but for n = 1 some exact answers have beenobtained.

Let 1 < p < ∞ and let ω : [a, b] → [0,∞) be a weight in L1(a, b). Set

Gp(ω) = supI

(∫−I

ωpdx

) 1

p

(∫−I

ω dx

)−1

where the supremum is taken over all intervals I ⊂ [a, b]. If Gp(ω) < ∞, then ω

is said to belong to the Gp-class of Gehring.

For a non-constant quasiminimizer u in [a, b] set

Kp(u) = sup[c,d]

d∫

c

|u′|p dx(d − c)p−1

|u(d) − u(c)|p

where the supremum is taken over all intervals [c, d] ⊂ [a, b]. In other words,Kp(u) is the least constant in (2.2). The following lemma is immediate.

Lemma 5.2. Let u : [a, b] → R be absolutely continuous and non-constant with

u′ ≥ 0 a.e. Then u is Kp(u)-quasiminimizer with exponent p, p > 1, if and only

if u′ belongs to the Gp-class with Gp(u

′) = Kp(u)1

p .

A. Korenovskii [K] has determined the optimal higher integrability bound p0 =p0(p,K) for a weight ω in the Gp-class. Hence Lemma 5.2 enables us to determi-nate the optimal integrability bound for the derivative of a Kp-quasiminimizer.Let γp,t : [p,∞) → R, p > 1, t > 1, be the function

γp,t(x) = 1 − tp x − p

x(

x

x − 1)p

,

and let p1(p, t) ∈ (p,∞) be the unique solution of the equation γp,t(x) = 0. Forthe properties of p1(p, t) see [DS, Section 3].

Theorem 5.3. Suppose that u is a (p,K)-quasiminimizer, p ≥ 1, K ≥ 1, in

[a, b]. Then u′ ∈ L

s(a, b) for 1 ≤ s < p1(p,K1

p ) if p > 1 and K > 1, u′ ∈ L

1(a, b)if p = 1 and u

′ ∈ L∞(a, b) if p > 1 and K = 1. All these integrability conditions

are sharp.


Proof. Let first p > 1 and K > 1. We may assume that u is increasing. By

Lemma 5.2, Gp(u′) = Kp(u)

1

p and from [K, Theorem 2] we conclude that u′ ∈

Ls(a, b) for 1 ≤ s < p1(p,Gp(u

′)) = p1(p,Kp(u)1

p ).

If p > 1 and K = 1, then u is affine and hence u′ ∈ L

∞(a, b). For p = 1, u′

trivially belongs to L1(a, b).

To see that the bound α = p1(p,K1

p ) is sharp for p > 1 and K > 1 it sufficesto consider the interval [0, 1]. The function

u(x) =α

α − 1x

α−1

α , x ∈ [0, 1],

has the derivative u′(x) = x

− 1

α and a direct computation shows that u′ belongs

to the Gehring class with Gp(u′) = K

1

p , see [DS, Proposition 2.3]. By Lemma 5.2u is a K-quasiminimizer. On the other hand, u

′ does not belong to Lα(0, 1). This

shows that the open ended upper bound p1(p,K1

p ) is sharp.

For p = 1 the integrability of u′ cannot be improved since every increasing

absolutely continuous function u is a 1-quasiminimizer. The theorem follows.

Remark 5.4. For p = 2,

p1(2, t) = 1 + t(t2 − 1)−1

2 , t > 1,

and hencep1(2, K

1

2 ) = 1 + K1

2 (K − 1)−1

2 , K > 1.

In [MS] the inverse functions of one dimensional quasiminimizers are alsoconsidered.

6. Quasisuperharmonic functions

In the nonlinear potential theory superharmonic functions can be defined inmany ways. The most natural definition uses the comparison principle, see (6.3)below. Let p > 1 and let Ω be an open set in R

n. A function u : Ω → (−∞,∞]is said to be superharmonic, i.e. (p, 1)-superharmonic, if

(6.1) u is lower semicontinuous,(6.2) u 6≡ ∞ in any component of Ω,

(6.3) for each open set Ω′ ⊂⊂ Ω and each minimizer h ∈ C(Ω′), i.e. (p, 1)-

quasiminimizer, the inequality u ≥ h on ∂Ω′ implies u ≥ h in Ω′.

For the theory of superharmonic functions in the nonlinear situation see [HKM].

Superharmonic functions can also be defined with the help of minimizers, see[HKM, Theorem 7.10] and [HKM, Corollary 7.20]. For other definitions see [B].

Lemma 6.4. Suppose that u : Ω → (−∞,∞] satisfies (6.1) and (6.2). Then

u is (p, 1)-superharmonic if and only if there is an increasing sequence (u∗i ) of

(p, 1)-quasisuperminimizers, i.e. superminimizers, with u = lim u∗i . Here u

∗i is

the lower semicontinuous representative of a superminimizer ui.


Note that given a superharmonic function u the sequence u∗i = min(u, i), i =

1, 2, . . ., is the required sequence.

In view of Lemma 6.4 the following definition for (p,K)-quasisuperharmonicityis natural.

Definition 6.5. Let Ω ⊂ Rn be an open set, p > 1 and K ≥ 1. A function

u : Ω → (−∞,∞] is said to be (p,K)-quasisuperharmonic if there is an increasingsequence of K-quasisuperminimizers ui in Ω such that lim u

∗i = u and u 6≡ ∞ in

each component of Ω.

Lemma 6.6. Suppose that u is a (p,K)-quasisuperharmonic function in Ω and

locally bounded above. Then u is a (p,K)-quasisuperminimizer.

Proof. By the definition for quasisuperharmonicity there is an increasing se-quence of quasisuperminimizers u

∗i : Ω → (−∞,∞) such that lim u

∗i = u. From

Theorem 4.14 it follows that u is a K-quasisuperminimizer as required.

Note that a (p,K)-quasisuperharmonic function is automatically lower semi-continuous as a limit of an increasing sequence of lower semicontinuous functions.

Lemma 6.7. Let u be a K-quasisuperharmonic function in Ω and h a (contin-

uous) minimizer in Ω. Then min(u, h) is a K-quasisuperminimizer (and hence

K-quasisuperharmonic) in Ω.

Proof. Let u∗i be an increasing sequence of K-quasisuperminimizers in Ω such

that u∗i → u. Now min(u∗

i , h) is lower semicontinuous and by Corollary 4.4,min(u∗

i , h) is a K-quasisuperminimizer. Since min(u∗i , h) ≤ h, it follows that

min(u, h) = lim min(u∗i , h) is a K-quasisuperharmonic function. By Lemma 6.6,

min(u, h) is a K-quasisuperminimizer.

Theorem 6.8. Suppose that u : Ω → (−∞,∞] satisfies (6.1) and (6.2). Then u

is a K-quasisuperharmonic if and only if min(u, c) is a K-quasisuperminimizer

for each c ∈ R.

Proof. The only if part follows from Lemma 6.7. For the sufficiency choose c =i, i = 1, 2, . . .. Then min(u, i) is a lower semicontinuous K-quasisuperminimizerand it follows from Definition 6.5 that u is a K-quasisuperharmonic function.

The theory for K-quasisuperharmonic functions is still in its infancy. However,the following result was proved in [KM2]: A set C ⊂ R

n is said to be (p,K)-polar,if there is a neighborhood Ω of C and a (p,K)-quasisuperharmonic function u inΩ such that u(x) = ∞ for each x ∈ C. Then C is a (p,K)-polar set if and onlyif the p-capacity of C is zero. It has been previously known that a set C ⊂ R

n

is a (p, 1)-polar set if and only if the p-capacity of C is zero. Hence allowing thefreedom due to K ≥ 1 adds nothing new to the structure of polar sets.

7. Appendix 1

Proof for Theorem 4.12. That P2(u, Ω′) is a subminimizer in Ω′ followsfrom Lemma 4.9. In order to show that w = P2(u, Ω′) is a K-quasisuperminimizer


in Ω let Ω′′ ⊂⊂ Ω be open and v a function such that v − w ∈ W1,p0

(Ω′′) andv ≥ w in Ω′′. We set v = w in Ω \Ω′′. For the K-quasisuperminimizing propertyof w it suffices to show

(a)

∫

Ω′′∩w<v

|∇w|pdx ≤ K

∫

Ω′′∩w<v

|∇v|pdx.

Hence we may assume that w < v in Ω′′ although Ω′′ ∩ w < v need not bean open set. Write A = x ∈ Ω : u(x) < v(x). Then A ⊂ Ω′′ because ifx ∈ A \ Ω′′, then u(x) < v(x) = w(x) which is a contradiction since u ≥ w. Thequasisuperminimizing property of u yields

(b)

∫

A

|∇u|pdx ≤ K

∫

A

|∇v|pdx,

see Remark 4.2 (c). The function min(u, v) satisfies w ≤ min(u, v) ≤ u in Ω andmin(u, v) can be continued as w to Ω′′ \ w < u. The resulting function is inthe right Sobolev space. Note also that min(u, v) = w outside Ω′′ ∩ Ω′ and thatmin(u, v) and w coincide outside w < u ∩ Ω′′ in Ω. Since w is the solutionto the K−

u (Ω′)-obstacle problem, w ≤ min(u, v) and min(u, v) has the correctboundary values w in w < u ∩ Ω′′, we obtain

(c)

∫

w<u∩Ω′′

|∇w|p dx ≤∫

w<u∩Ω′′

|∇min(u, v)|p dx

=

∫

w<u∩Ω′′∩u<v

|∇u|p dx +

∫

w<u∩Ω′′∩u≥v

|∇v|p dx.

Since

(Ω′′ ∩ w = u) ∪ (w < u ∩ Ω′′ ∩ u < v) ⊂ Ω′′ ∩ u < v,the inequalities (b) and (c) yield∫

Ω′′

|∇w|pdx =

∫

Ω′′∩w=u

|∇u|p dx +

∫

Ω′′∩w<u

|∇w|p dx

≤∫

Ω′′∩w=u

|∇u|p dx +

∫

w<u∩Ω′′∩u<v

|∇u|p dx

+

∫


|∇v|p dx

≤∫

Ω′′∩u<v

|∇u|p dx +

∫


|∇v|p dx

≤ K

∫

Ω′′∩u<v

|∇v|p dx +

∫


|∇v|p dx ≤ K

∫

Ω′′

|∇v|p dx.


This is (a). We leave to an exercise to show that w is a K-quasiminimizer in Ω′.The proof is complete.

8. Appendix 2

Proof for Theorem 4.14. We show that the quasisuperminimizing propertyis preserved under the increasing convergence if the limit is locally bounded aboveor belongs to W

1,ploc

(Ω).

The proof of this theorem [KM2, Theorem 6.1] contains a gap which will besettled here. The argument is quite similar as in [KM2]. The authors would liketo thank professor Fumi–Yuki Maeda for pointing out the error in the originalpaper.

We consider the case (i) only. The case (ii) follows from (i) and from an easytruncation argument, see [KM2, p. 477]. In the case (i) it follows from the DeGiorgi type upper bound

∫

B(x,ρ)

|∇ui|p dx ≤ c(R − ρ)−p

∫

B(x,R)

(ui − k)pdx,

where

k < − supess supB(x,R)

ui : i = 1, 2, . . .,

0 < ρ < R and B(x,R) ⊂⊂ Ω, that the sequence (|∇ui|) is uniformly boundedin L

p(Ω′) for every Ω′ ⊂⊂ Ω. This implies that u ∈ W1,ploc

(Ω) and we may assumethat (|∇ui|) converges weakly to ∇u in L

p(Ω′).

Let C ⊂ Ω be a compact set and for t > 0 write

C(t) = x ∈ Ω : dist(x,C) < t.

Then C(t) ⊂⊂ Ω for 0 < t < dist(C, ∂Ω) = t0.

Lemma 8.1. Let u and ui be as in Theorem 4.12. Then for almost every t ∈(0, t0) we have

lim supi→∞

∫

C(t)

|∇ui|p dx ≤ c

∫

C(t)

|∇u|p dx

where the constant c depends only on K and p.

Proof. Let 0 < t′< t < t0 and choose a Lipschitz cut-off function η such that

0 ≤ η ≤ 1, η = 0 in Ω \ C(t) and η = 1 in C(t′). Let

wi = ui + η(u − ui), i = 1, 2, . . . .


Then wi − ui ∈ W1,p0

(C(t)) and wi ≥ ui a.e. in C(t). Hence the quasisupermini-mizing property of ui gives

∫

C(t′)

|∇ui|p dx ≤∫

C(t)

|∇ui|p dx ≤ K

∫

C(t)

|∇wi|p dx

≤ αK

(∫

C(t)

(1 − η)p|∇ui|p dx

+

∫

C(t)

|∇η|p(u − ui)pdx +

∫

C(t)

ηp|∇u|p dx

),

where α = 2p−1. Adding the term

αK

∫

C(t′)

|∇ui|p dx

to the both sides and taking into account that η = 1 in C(t′) we obtain

(1 + αK)

∫

C(t′)

|∇ui|p dx ≤ αK

∫

C(t)

|∇ui|p dx

+αK

∫

C(t)

|∇η|p(u − ui)pdx + αK

∫

C(t)

|∇u|p dx.

Set Ψ : (0, t0) → R,

Ψ(t) = lim supi→∞

∫

C(t)

|∇ui|p dx.

Now −ui belongs to the De Giorgi class (see [KM2, Lemma 5.1]), and hence Ψis a finite valued and increasing function of t. Hence the points of discontinuityform a countable set. Let t, 0 < t < t0, be a point of continuity of Ψ. Lettingi → ∞, we obtain from the previous inequality the estimate

(1 + αK)Ψ(t′) ≤ αKΨ(t) + αK

∫

C(t)

|∇u|p dx,

because ∫

C(t)

|∇η|p(u − ui)pdx → 0

as i → ∞ by the Lebesgue monotone convergence theorem. Since t is a point ofcontinuity of Ψ, we conclude that

(1 + αK)Ψ(t) ≤ αKΨ(t) + αK

∫

C(t)

|∇u|p dx,


or in other words

Ψ(t) ≤ αK

∫

C(t)

|∇u|p dx.

Proof of Theorem 4.14, case (i). As noted before u ∈ W1,ploc

(Ω). Let Ω′ ⊂⊂ Ω

be open and v ∈ W1,p(Ω′), v ≥ u almost everywhere and v − u ∈ W

1,p0

(Ω′). By[KM2, Lemma 6.2] it suffices to show that

(a)

∫

Ω′

|∇u|p dx ≤∫

Ω′

|∇v|p dx.

To this end let ε > 0 and choose open sets Ω′′ and Ω0 such that

Ω′ ⊂⊂ Ω′′ ⊂⊂ Ω0 ⊂⊂ Ω

and

(b)

∫

Ω0\Ω′

|∇u|p dx < ε.

Next choose a Lipschitz cut-off function η with the properties η = 1 in a neigh-borhood of Ω′, 0 ≤ η ≤ 1 and η = 0 on Ω \ Ω′′. Set

wi = ui + η(v − ui), i = 1, 2, . . . .

Then wi − ui ∈ W1,p0

(Ω′′) and wi ≥ ui. Thus(∫

Ω′′

|∇wi|p dx

)1/p

≤(∫

Ω′′

((1 − η)|∇ui|p + η|∇v|)pdx

)1/p

+

(∫

Ω′′

|∇η|p(v − ui)pdx

)1/p

= αi + βi.

The elementary inequality

(αi + βi)p ≤ α

pi + pβi(αi + βi)

p−1

implies that

(c)

∫

Ω′′

|∇wi|p dx ≤∫

Ω′′

(1 − η)|∇ui|p dx +

∫

Ω′′

η|∇v|p dx + pβi(αi + βi)p−1

,

where we also used the convexity of the function t 7→ tp. We estimate the terms

on the right-hand side separately.

Since η = 1 in a neighborhood of Ω′, there is a compact set C ⊂ Ω′′ such thatC ∩ Ω′ = ∅ and ∫

Ω′′

(1 − η)|∇ui|p dx ≤∫

C

|∇ui|p dx.


We can choose C = Ω′′ \ Ω′(t) for sufficiently small t > 0. Next choose t > 0such that

lim supi→∞

∫

C(t)

|∇ui|p dx ≤ c

∫

C(t)

|∇u|p dx

and C(t) ⊂ Ω0 \ Ω′. This is possible by Lemma 8.1. We have

lim supt→∞

∫

Ω′′

(1 − η)|∇ui|p dx ≤ lim supi→∞

∫

C

|∇ui|p dx

≤ lim supi→∞

∫

C(t)

|∇ui|p dx ≤ c

∫

C(t)

|∇u|p dx ≤ cε

where the last inequality follows from (b). Since ∇η = 0 in Ω′ and v = u inΩ′′ \ Ω′ the Lebesgue monotone convergence theorem yields βi → 0 as i → ∞.On the other hand the numbers αi remain bounded as i → ∞. Hence it followsfrom (c) that

lim supi→∞

∫

Ω′′

|∇wi|p dx ≤ cε +

∫

Ω′′

η|∇v|p dx ≤ cε +

∫

Ω′′

|∇v|p dx.

Now ui is a K-quasisuperminimizer and hence for large i it follows that∫

Ω′

|∇ui|p dx ≤∫

Ω′′

|∇ui|p dx ≤ K

∫

Ω′′

|∇wi|p dx ≤ 2Kcε + K

∫

Ω′′

|∇v|p dx

≤ 2Kcε + K

∫

Ω′

|∇v|p dx + K

∫

Ω′′\Ω′

|∇v|p dx

≤ 3Kcε + K

∫

Ω′

|∇v|p dx,

where we used (b) and the fact that ∇u = ∇v in Ω′′ \ Ω′. Since ε > 0 wasarbitrary and since ∫

Ω′

|∇u|p dx ≤ lim infi→∞

∫

Ω′

|∇ui|p dx

by the weak convergence ∇ui → ∇u in Lp(Ω′), this completes the proof of (a)

and the proof for the case (i) is complete.

References

[B] Bjorn, A., Characterization of p-superharmonic functions on metric spaces, StudiaMath. 169 (1) (2005), 45-62.

[BB] Bjorn, A. and J. Bjorn, Boundary regularity for p-harmonic functions and solutionsof the obstacle problem, preprint, Linkoping, 2004.

[BBS] Bjorn, A., J. Bjorn and N. Shanmugalingam, The Perron method for p-harmonicfunctions, J. Differential Equations 195 (2003), 398-429.


[BJ1] Bjorn, J., Poincare inequalities for powers and products of admissible weights, Ann.Acad. Sci. Fenn. Math. 26 (2001), 175-188.

[BJ2] Bjorn, J., Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math.46 (2002), 383-403.

[DS] D’Apuzzo, L. and C. Sbordone, Reverse Holder inequalities. A sharp result, Rend.Matem. Ser. VII, 10 (1990), 357-366.

[DT] Di Benedetto, E. and N.S. Trudinger, Harnack inequalities for quasi-minima of vari-ational integrals, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 295-308.

[GG1] Giaquinta, M. and E. Giusti, On the regularity of the minima of variational integrals,Acta Math. 148, (1982), 31-46.

[GG2] Giaquinta, M. and E. Giusti, Qasiminima, Ann. Inst. H. Poincare Anal. Non Lineaire1 (1984), 79-104.

[G] Giaquinta, M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic

Systems, Ann. of Math. Studies 105, Princeton Univ. Press, 1983.[HKM] Heinonen, J., T. Kilpelainen and O. Martio: Nonlinear Potential Theory of Degener-

ate Elliptic Equations, Oxford University Press, 1993.[KM1] Kinnunen, J. and O. Martio: Nonlinear potential theory on metric spaces, Illinois J.

Math. 46 (2002), 857-883.[KM2] Kinnunen, J. and O. Martio, Potential theory of quasiminimizers, Ann. Acad. Sci.

Fenn. Math. 28 (2003), 459-490.[KS1] Kinnunen, J. and N. Shanmugalingam, Regularity of quasi-minimizers on metric

spaces, Manuscripta Math. 105 (2001), 401-423.[KS2] Kinnunen, J. and N. Shanmugalingam, Polar sets on metric spaces, Trans. Amer.

Math. Soc. 358.1 (2005), 11-37.[KSa] Krylov, N.V. and M.V. Safalow, Certain properties of solutions of parabolic equations

with measurable coefficients (Russian), Izv. Akad. Nauk. SSSR 40 (1980), 161-175.[K] Korenovskii, A.A., The exact continuation for a reverse Holder inequality and Muck-

enhoupt’s conditions, Transl. from Matem. Zametki, 52(6) (1992), 32-44.[Ma] Marola, N., Moser’s method for minimizers on metric measure spaces, Helsinki Uni-

versity of Technology, Institute of Mathematics, Research Reports A 478, 2004.[MS] Martio, O. and C. Sbordone, Quasiminimizers in one dimension – Integrability of the

derivative, inverse function and obstacle problems, Ann. Mat. Pura Appl., to appear.[Sh1] Shanmugalingam, N., Newtonian spaces: An extension of Sobolev spaces to metric

measure spaces, Revista Math. Iberoamericana 16 (2000), 243-279.[Sh2] Shanmugalingam, N., Harmonic functions on metric spaces, Illinois J. Math. 45

(2001), 1021-1050.[Sh3] Shanmugalingam, N., Some convergence results for p-harmonic functions on metric

measure spaces, Proc. London Math. Soc. 87 (2003), 226-246.

Olli Martio E-mail: [email protected]: Department of Mathematics and Statistics, BOX 68, FI-00014 University of Helsinki,

FINLAND


History and Recent Developments in Techniques forNumerical Conformal Mapping

R. Michael Porter

Abstract. A brief outline is given of some of the main historical develop-ments in the theory and practice of conformal mappings. Originating with thescience of cartography, conformal mappings has given rise to many highly so-phisticated methods. We emphasize the principles of mathematical discoveryinvolved in the development of numerical methods, through several examples.

Keywords. numerical conformal mapping, cartography, osculation, interpo-lating polynomial method, mathematical discovery.

2000 MSC. 65-03 30C30 65-02 65S05.

Contents

1. Introduction 208

2. A Brief History of Mapmaking 208

3. The General Problem of Conformal Mappings 214

4. The Crowding Problem 216

5. Elementary Facts about Analytic Functions 217

6. Osculation Methods 218

6.1. Koebe’s method. 218

6.2. Graphical methods. 219

6.3. Grassmann’s method. 220

6.4. Sinh-log method 220

7. Schwarz-Christoffel Methods 222

8. Rapidly Converging Methods 223

8.1. Theodorsen’s Method. 223

8.2. Fornberg’s method. 225

9. Generalities on Conformal Mapping Methods 226

10. Interpolating Polynomial Method 226

10.1. Some properties of the half-click mapping. 228

10.2. Interpolating polynomial algorithm 229

Research supported by CONACyt grant 46936.

208 R. Michael Porter IWQCMA05

10.3. Numerical example 230

11. The Quest for Better Methods 233

11.1. Methods using derivatives 233

11.2. Method of simultaneous interpolation 233

11.3. Minimization approach 235

12. Combined Methods 235

13. Epilogue 236

References 237

1. Introduction

Although the present meeting is mainly concerned with the study of the the-ory of quasiconformal mapping, our topic here is to understand some of the basicprinciples of the theory of conformal mapping, with emphasis on the computa-tional perspective.

The history of quasiconformal mappings is usually traced back to the early1800’s with a solution by C. F. Gauss to a problem which will be briefly mentionedat the end of Section 2, while conformal mapping goes back to the ideas of G.Mercator in the 16th century. Since the early work had much to do with theproduction of maps of the physical world, we will begin with a brief survey ofhow this came about.

2. A Brief History of Mapmaking

From antiquity mankind has needed to map out his world: whether for con-trolling dominated territories or to travel great distances. For the moment, bymap we will mean a representation of part of the earth’s surface on a flat paper.Indeed, as one historian writes, mapmaking is older than the written word:

“The human activity of graphically translating one’s perception of hisworld is now generally recognized as a universally acquired skill and onethat pre-dates virtually all other forms of written communication.”1

A map obviously must contain “known reference points” and somehow showtheir relative distances and directions. This is the basis of the humor in thefollowing lines from “The Hunting of the Snark” by Lewis Carroll,

1www.henry-davis.com/MAPS/AncientWebPages/100mono.html. As with many of the webreferences cited here, the same information may be found on many sites and it is difficult topinpoint an original source.

Techniques for Numerical Conformal Mapping 209

Figure 1. “Map” lacking reference points

He had bought a large map2 representing the sea,Without the least vestige of land:And the crew were much pleased when they found it to beA map they could all understand.

“What’s the good of Mercator’s North Poles and Equators,Tropics, Zones, and Meridian Lines?”So the Bellman would cry: and the crew would reply“They are merely conventional signs!”

“Other maps are such shapes, with their islands and capes!But we’ve got our brave Captain to thank”(So the crew would protest) “that he’s bought us the bestA perfect and absolute blank!”

2http://www.eq5.net/carrol/fit2.html

Figure 2. Left: Perspective view of terrain. Right: Topographi-cal map of same terrain.


Figure 3. Right: Reconstruction of Catalhoyuk map, which issaid to have had the function of registering property ownership.

The picture of a terrain in Figure 2 (left) is likewise not a map. In the map ofFigure 2 (right), the reference points are “subjective,” involving loci of constantheight which need not correspond to any physically noticeable characteristics ofthe terrain. Maps may incorporate our preconceptions or prejudices of what theworld looks like.

Of course, in modern mathematics the matter of reference points is resolvedby the precise notion “function,” for which it is postulated that every point ofthe surface is made to correspond, albeit theoretically, to a unique point of themap.

Let us take a quick trip though the history of maps. A nine-foot-long stonemap dating from 6200 B.C., which appears to be a plan of a town predatingAnkara, Turkey3, is shown in Figure 3. It is said to have served to registerproperty rights, perhaps for tax purposes.

Moving now to larger-scale maps, we have a map of Africa produced in silk,in 1389 by the Chinese map Great Ming Empire), measuring 17 square meters(Figure 4). It is on display in South Africa and is said to be a copy of an earlierstone one4. However, it would be a mistake to say that the larger the scale, themore recent the map. In 1999 there was discovered in Ireland a map dating from

3John F. Brock, “The Oldest Cadastral Plan Ever Found: the Catalhoyuk town plan of6200 B.C.,” http://www.mash.org.au/articles/articles2.htm

4BBC News, http://news.bbc.co.uk/2/hi/africa/2446907.stm

Figure 4. Chinese map of Africa


Figure 5. Alleged ancient moon map

about 3000 B.C. which is claimed to be the oldest known map of the moon.5

Even older are the well-known drawings in from 14,000 B.C the Lascaux Caves,France, discovered in 1940, with a drawing of the constellation Pleiades and ofthe “Summer Triangle,” and a map of the constellation Orion from 30,000 B.C.found engraved on a 4 cm. ivory tablet in Germany in 1979.6

At the time of this writing, it is being investigated whether or not the Chineseadmiral Zheng He discovered America and circled the globe 80 years before thevoyages of Christopher Columbus; part of the argument is based on a map fromthe year 1418.7 At any rate, it is often stated incorrectly that in the epoch ofColumbus, people generally thought that the earth was flat. While the sphericalnature of the earth’s surface was known to the ancient Greeks, and its diameterwas measured by Hipparcus (190–120 A.C.), it is significant that no early mapwas produced similar to that of Figure 6.

The principle here is that consciously or not, one tends to look for the simplestpossible answers to mathematical questions. Here it would be the topology whichis unnecessarily complicated. The possibility of closed curves on the earth’s

5http://news.bbc.co.uk/1/hi/sci/tech/1205638.stm6http://news.bbc.co.uk/1/hi/sci/tech/871930.stm and http://news.bbc.co.uk/1/

hi/sci/tech/2679675.stm7The Sunday Statesman, New Delhi, 15 January, 2006; http://edition.cnn.com/2003

/SHOWBIZ/books/01/13/1421/index.html

Figure 6.


Figure 7. If there were an isometry φ from a neighborhood ofthe North Pole to a planar region, then circles of radius r in thetwo spaces would have different circumferences.

surface which are not contractible to a point would have many consequences,among them a nonconstant curvature of the surface (at least according to ourintuitive notions of Euclidean space).

So let us return to the idea of a spherical earth. The oldest known maps of thecelestial sphere were not on flat surfaces such as paper, but rather on tortoiseshells. One reason for this may have been that, as Figure 7 shows, there can

be no isometry (that is, a distance-preserving map) from a spherical region to a

planar one.

This statement may sound disappointing, because it would be extremely usefulto be able to determine one’s position on a map using the distance one hastraveled from a previously determined point. However, navigators of the MiddleAges realized that if distances cannot be preserved on planar maps of sphericalsurfaces, then at least it would be useful to conserve angles. When the anglesbetween curves on the earth are equal to the corresponding angles on the map, themap is said to be conformal (Figure 8). A navigator on the high seas, orienting

Figure 8. Conformal mapping of plane domains and of surfaces.


himself by the stars, has to go through some rather involved calculations todetermine the distance he has just traveled. However, the stars tell him quicklythe direction in which he is traveling, and hence the angle which his course ismaking with lines of latitude and longitude. Thus a conformal (nonisometric)map is not necessarily as impractical as it may first seem.

As is mentioned also in A. Rasila’s article, the idea of a conformal earth mapwas developed by G. Mercator. The map is designed in two steps. The firststep is to project all points of the sphere except the poles to a cylinder in whichthe sphere is inscribed (Figure 9). Then the horizontal lines in the image areto be adjusted, placing the line corresponding to latitude θ at the height ϕ(θ).Mercator did not know how to do this second step precisely; it was E. Wright in1599 who found the theoretical solution,

ϕ(θ) =

∫ θ

0

dθ

cos θ.

At that time no one knew how to evaluate this integral in elementary terms.This led to the compilation of numerical tables for navigators, which was carriedout by successively summing small increments of the integrand. After some time,people compared different tables and observed a curious coincidence: the integral

∫ θ

0

dθ

cos θ

-

?

(1)

(2)

Figure 9. Steps in Mercator projection


was apparently approximately equal to

log tan(

θ

2+

π

4

),

a statement which became a conjecture and was finally proved in 1668 by J. Gre-gory.

The point of this rather long digression is to illustrate that the interplaybetween “pure” and “applied” mathematics has been with us for a long time.Numerical evidence has long played an important role in the generation of con-jectures and theorems; leaving aside questions arising from ancient geodesy, wemention only the Prime Number Theorem conjectured by Gauss, and the locationof zeroes of the zeta function known as the Riemann Hypothesis.

Let us look again at Figure 8. The problem of finding a conformal correspon-dence between an arbitrary region and a plane region is rather complicated, andclassically is called the problem of finding isothermic coordinates for a surface.For real-analytic surfaces, this was solved by Gauss. The solution requires quasi-conformal mappings as an intermediate step, and will not be explained in detailhere.

3. The General Problem of Conformal Mappings

Here we limit the discussion to plane domains. From the theory of functionsof a Complex Variable we have the following basic fact.

A correspondence f between plane domains is conformal if and only

if it can be represented locally as an analytic function of a complex

variable z = x + iy,

f(z) =∞∑

0

ak(z − z0)k.

Thus, once a conformal mapping is found from a given surface to a planeregion, all other conformal mappings are obtained from this one by composingwith an analytic function.

Applications of conformal mapping to physics, too numerous to mention here,require finding conformal mappings between two given domains. For the simplestsituation, that of simply connected domains, the problem of finding a singlemapping is clarified by the following well-known result.

Theorem 1. RIEMANN MAPPING THEOREM: There exists a conformal map-

ping f from the unit disk D = z : |z| < 1 to any simply connected proper

subdomain D of the complex plane (it is unique if suitably normalized).

Knowing that a conformal mapping exists is not the same as knowing howto solve the numerical problem: given D, to calculate f to a given degree ofaccuracy. In some simple cases the conformal mapping can be written as acomposition of elementary functions, such as Mobius transformations, powers


and roots, trigonometric functions, elliptic integrals, and the like. These basicanalytic formulas are also useful for reducing general problems to more accessibleones (see Section 6.2 below). In practice, a mapping problem can be presented ina variety of ways, partly because a domain can be presented in a variety of ways:as set of points satisfying a certain condition (such as an inequality F (z) < c), ora specific list of points (such as a set of pixels in an image). More commonly, adomain is specified in terms of its boundary, which may be a condition satisfiedby its points (such as an equality F (z) = c), or a parametrization z = γ(t), orsimply a list of points along the boundary.

Furthermore, a map from one domain D1 to D2 is often described as a com-position of a map from D to D1 composed with the inverse of a map from D toD2. Depending on the nature of the numerical description, it may be difficult orinconvenient to calculate the inverse map.

The following result is also relevant to numerical work.

Theorem 2. (Caratheodory) Every conformal mapping between to Jordan re-

gions extends to a homeomorphism of the closures of the regions.

Because of the Cauchy Integral Formula

(1) f(z0) =

∫

∂D

f(z)

z − z0

dz, z0 ∈ D,

which is valid for analytic functions in D continuous on the boundary, it is

sufficient to know the values of a conformal mapping on the boundary of D. Thisobservation reduces the complexity of the conformal mapping problem from 2

Given data:

(i) D (or ∂D)(ii) z0 ∈ D

Figure 10. Illustration by G. Francis in [1]. Equipotential linesof ideal fluid flow correspond to concentric circles under Riemannmapping.


dimensions to 1, and this is why the vast majority of the numerical methodswhich have been developed focus on calculating the boundary mapping f |

∂D. A

numerical scheme for evaluating (1), once the boundary mapping is known, canbe found in [12].

For this reason, we will consider the problem as defined in Figure 11 (exceptfor Section 6, where the inverse mapping will be sought). The closed curvet 7→ γ(t) defines the boundary of D. A point z0 fixed inside of γ is to be equalto f(0). We are interested in the boundary values f(eiθ) for 0 ≤ θ < 2π. Themapping θ 7→ e

iθ of the interval to the unit circumference is so natural that weuse it in the statement of the problem: to find a function t = b(θ) so that

(2) f(eiθ) = γ(b(θ)).

0 2π

γ

t

0 2π

θ

e

iθb

Figure 11. Elements of mapping problem (picture taken from [8])

4. The Crowding Problem

When f is a conformal mapping, the value of |f ′| (or the density of the gridpoints required) may vary greatly along different parts of the boundary. Thecrowding factor is the ratio of the greatest to the least value of |f ′|. For exam-ple, consider conformal maps fa from D (or from a square, as in Figure 12) torectangles of height 1 and base a > 0, with fa(0) at the center of the rectangle.Then the crowding factor grows exponentially with a as a → ∞.

This creates a numerical difficulty. Suppose we have calculated b(2πj/N) forj = 1, . . . , N , that is, for N equally spaced values of θ. If the crowding factor


Figure 12. Conformal maps between non-similar rectangles can-not send vertices to vertices.

is very large, then many of these values will be bunched together, while otherpoints will be spread relatively far apart. This may cause an imperfect pictureof the behavior the mapping function (see for example the last picture in Figure29 below). The crowding factor for the domain in Figure 11 is approximately1000.

5. Elementary Facts about Analytic Functions

There are a great many theorems in Complex Analysis of the form “the func-tion f is analytic if and only if . . . ”. A conformal mapping is the same thing asa 1-to-1 analytic function, and this notion also admits many characterizations.

Theorem 3. An analytic and 1-to-1 function f : D → D is onto D when |f ′(z0)|is maximal among the class of injective analytic functions D → D sending the

point z0 to 0.

Likewise there are many characterizations of boundary values on ∂D = |z| =1 for conformal mappings. We list without detailed explanations, a few whichhave been used as the basis for numerical methods.

Theorem 4. A continuous function γ : ∂D → C is the boundary value of an

analytic function if:

1. [12] K[Re γ] = Im γ, where K is the conjugate boundary operator;

or if

2. [8] its Fourier series γ(t) =∑∞

−∞ bkekit has bk = 0 for k < 0;

or if

3. [17],[22] |γ′(t)|1/2S(γ(t), z0) +

∫ β

0|γ′(t)|1/2

A(γ(t), γ(s))|γ′(s)|1/2

= |γ′(t)|1/2H(z0, γ(t)),

where A and H are the Kerzman-Stein and the Cauchy kernels;

or if

4. etc., etc.

Generally speaking, we may say that each theoretical characterization of con-formal mappings can lead to a numerical method or to a family of numericalmethods. At the risk of oversimplifying, we can say that most methods fall intoone of the following two classes.

• “Easy Methods”: D → D

• “Fast Methods”: D → D


1. 2.

3. 4.

× ×

××

Figure 13. One iteration of the Koebe square-root method

We give some illustrations below.

6. Osculation Methods

Most of the “Easy Methods” are classified as osculation methods, which consistof first mapping D into D, and then mapping the image region to a largerregion inside of D, and so on. The desired approximation to f : D → D is thecomposition of these mappings, and f itself is their limit.

6.1. Koebe’s method. The first osculation method ever created is based onthe proof given by P. Koebe in 1905 for Theorem 3, and which is found in manyComplex Variables texts, such as [2], [5]. This procedure is illustrated in Figure13. The steps are prescribed as follows.

0. Suppose D ⊂ D

1. Find t0 with |γ(t0)| < 1 minimal (“worstboundary point”). Let a0 = γ(t0)

2. Move a0 to 0: w1 =w − a0

1 − a0w

3. Take square root: w2 =√

w1

4. Move the image b0 of zero back to 0:

w =w2 − b0

1 − b0w2

The auxiliary mappings defined in steps 2, 3, and 4 combine to give a mapfrom D to itself which fixes the origin and has derivative greater than 1. Thus


the image under each successive iteration is “larger” than the previous one. Notethat if D = D, then step 2 cannot be applied.

The images of a domain under successive iterations of the Koebe method areshown in Figure 14. The convergence to ∂D can be made slightly faster by usingthe cube root instead of the square root in Step 3. If one uses the nth root andlets n → ∞, one approaches the Koebe logarithm method, in which the logarithmreplaces the nth root and step 4 is modified accordingly.

6.2. Graphical methods. In the 1950’s, people were looking for ways to find amapping of a given domain to a domain reasonably close to D, in order to applythen a fast method (such as Teodorsen’s method below) to the result. Today itseems incredible that this was done by hand. For example, in [11] a method wasdescribed by which the operator first fits manually the given domain D into adisk from which a sector bounded by two arcs has been removed (Figure 15). Aconformal mapping from this slightly larger domain D1 ⊃ D to D can be writtenin elementary terms,

(3)c + z1

c − z1

= k

(c + z

c − z

)1/α

.

as a composition of a power function with two Mobius transformations. TheseMobius transformations make the points 0,∞ correspond to key points in theoriginal figure. An ingenious scheme was presented in which one places thedomain over specially designed “graph paper” showing circles passing through

Figure 14. Iterations of the Koebe method (left) and the loga-rithmic Koebe method (right)


these two points and circles orthogonal to them. Then, much in the way thatone can magnify a picture by tracing a grid over it and then copying the part ofthe picture in each small square in it to the corresponding square in a larger grid,pieces of the boundary of the domain are copied to corresponding pieces within asecond graph paper. In this way the conformal mapping is approximated withoutcomputing the elements of (3) numerically.

6.3. Grassmann’s method. In 1979, E. Grassmann [10] automated and re-fined this idea of Albrecht-Heinhold, giving a much faster osculation method. Thefirst step is to detect automatically (that is, by a computer program) whetheran auxiliary mapping which opens a slit can be applied, and if not, applies otheralternatives. In the worst case, the Koebe method is used (Figure 16). The ideabehind this method is that for many domains, after a few iterations the domain is“opened up” sufficiently so that the better auxiliary mappings will be applicable.

6.4. Sinh-log method. This method is based on the observation [19] that inthe logarithmic Koebe method, at a certain step during each iteration, the log-arithmic image of the domain generally not only lies in the left half-plane, butwithin some horizontal band; that is, the imaginary parts of all its points arebounded above and below. An appropriate real-affine transformation takes thishalf-band to the normalized half-band −π/2 < Im z < π/2, which in turn thehyperbolic sine function sends to the whole left half-plane, as shown in Figure 17.As a result, a much larger part of the half-plane is covered before the half-planeis mapped back to the unit disk in the last step of the iterative procedure.

Figure 15. Manual graphical approach of Albrecht and Heinhold


Figure 16. Application of Grassmann’s method to a “daisy”domain. Each figure shows the slit which is opened to form thefollowing one.

Thus the “sinh-log” method is defined by following steps 0, 1, and 2 of theKoebe square-root algorithm, then (3) taking the logarithm of the resulting im-age; (4) applying an affine transformation to a subdomain of the normalizedhalf-band; (5) applying sinh, and finally (6) a Mobius transformation of the lefthalf-plane to D so that the composition of all the maps mentioned fixes theorigin.

The sinh-log algorithm converges roughly as does Grassmann’s for many stan-dard domains and for highly irregular domains it performs much better, whilebeing easier to program and more stable numerically. In particular, if one isinterested in calculating the composed mapping f and its inverse, it is easier tosave the data for a sequence of elementary maps of a single kind.

DG

Figure 17. One iteration of the sinh-log method. The lower leftpicture shows the half-strip containing the logarithmic image, aswell as the image of this under sinh


Figure 18. Iterations of the sinh-log method

It was mentioned that osculations are very “slow” methods. Their great ad-vantage is that they apply to any domain bounded by a continuous (say piecewisesmooth) closed curve. In general, osculation methods have the following charac-teristics.

Osculation Methods

COST OF ONE ITERATION: O(N)

RATE OF CONVERGENCE: very slow

GENERALITY: total

7. Schwarz-Christoffel Methods

The Schwarz-Christoffel methods are applicable to the particular case of adomain with polygonal boundary. Although for numerical work every domaincan be considered in principle “polygonal,” if the number of vertices is extremelylarge the advantage of Schwarz-Christoffel methods would be lost. The Schwarz-Christoffel formula says that the Riemann mapping from the unit disk to apolygonal domain is equal to the integral

(4) f(z) =

∫ z

0

dz

(z − z1)1−α1/π(z − z2)1−α2/π · · · (z − zn)1−αn/π.


w1

α1

w2

w3

Figure 19

For a given polygonal domain, the vertices wj = f(zj) are known as well asare the interior angles παj, but the prevertices zj are not known, so formula (4)is useless until these values are determined. A strategy for a typical Schwarz-Christoffel method would thus be

• guess approximate values for z1, z2, ...,• evaluate the Schwarz-Christoffel integral,• compare the results with w1, w2, ...,• apply some type of correction

The difficulty is to make an initial guess sufficiently close for this to work.A history and detailed explanation of various methods can be found in [6]. Anelegant solution to the Schwarz-Christoffel mapping question was invented byT. A. Driscoll and S. A. Vavasis [7], which begins by triangulating the polygonaldomain in a special way, and then solving a set of equations for the cross ratios ofall rectangles formed by pairs of adjacent triangles. Not only is there no problemto find an appropriate initial guess, but also the invariance of the cross ratiomakes it possible to avoid the crowding phenomenon: one can apply a Mobiustransformation of D to bring any part of the domain into focus.

8. Rapidly Converging Methods

It must be stressed that we will mention only a few of the very many methodswhich have been developed. Details and history of many classical methods willbe found in [12]; for more methods see Section 13.

8.1. Theodorsen’s Method. This method, presented in 1931 for improvingthe design of aircraft wings, applies only to starshaped domains (λz is in D

when z is in D and 0 < λ < 1). The region exterior to an airplane wing, inwhich the physics of air movement takes place, is obviously not of this form, butcan be reduced to it by a suitable auxiliary transformation (Joukowski profile,Figure 21).

So we assume from the start that the boundary ∂D is traced out by

(5) γ(t) = ρ(t)eit, 0 ≤ t ≤ 2π.


The idea behind the algorithm is that of condition 1 in Theorem 4. Considerthe real and imaginary parts u,v of the analytic function f(z) = u(z) + iv(z).They are known to be harmonic conjugates. Now if we are given a function u0

defined only on ∂D, it can be extended (by the Poisson integral) to a harmonicfunction u in all of D. This function has a harmonic conjugate v defined in ∂D.The restriction v0 of v to ∂D is called the conjugate boundary function of u0, andis written

(6) K[u0] = v0.

Now we apply the following facts.

1. The conjugate boundary function can be calculated by a singular integral

K[v0](eit) =

∫2π

0

v0(eis) cot

t − s

2ds.

2. If γ(b(θ)) = u0(eiθ) + iv0(e

iθ) and if v0 = K[u0], then γ b defines theboundary values of a conformal map.

Recall from Section 3 that we want t = b(θ) such that γ(b(θ)) defines theboundary values of an analytic function according to (2). This is equivalent to

b(θ) − θ = K[log(ρ b)](θ).

The difference δ(θ) = b(θ) − θ satisfies

(7) δ(θ) = K[log(ρ(θ + δ(θ))].

Thus δ is a fixed point of a nonlinear operator, and Theodorsen’s method saysto construct a sequence of functions by iterating it,

δk+1(θ) = K[ log(ρ(θ + δk(θ))) ].

Then δk → δ as k → ∞. The solution b is now given by b(θ) = θ + δ(θ).

Figure 20. Facsimile of Theodorsen’s Naca internal report


-1 -0.5 0.5 1

-1

-0.5

0.5

1

Figure 21. Joukowski profile and starshaped domain

Theodorsen’s Method

COST OF ONE INTERATION: O(N2)

RATE OF CONVERGENCE: linear

GENERALITY: starshaped domain;

close initial guess

8.2. Fornberg’s method. This method, published in 1980 in [8], is based oncriterion 2 of Theorem 4. We have as data for the problem γ : [0, 2π] → ∂D, aperiodic complex-valued function, together with a point z0 inside of D. We wantto find the boundary values γ b of the conformal mapping from D to D sending0 to z0. Like any map of the circle, it must have a Fourier series:

(8) γ(b(t)) =∞∑

k=−∞

bkekit

.

From what we have said γ(b(θ)) is to give the values on ∂D of an analytic functionf(z) =

∑∞k=0

akzk, so by (2)

(9)∞∑

k=0

ak ekiθ = f(eiθ) = γ(b(θ)) =

∞∑

k=−∞

bk ekiθ

and therefore

bk =

0, k < 0,ak, k ≥ 0.

This suggests the following procedure.


0. Guess initial values t1, t2, . . . , tN to be assigned as b(2jπ/N) =tj, i.e., we hope that f(e2jπi/N) = γ(tj) approximately.

1. For b determined this way, calculate the Fourier coefficients forγ b:

b−N , b−(N−1), . . . , b−1

2. Calculate the corresponding changes in bk which would go withslightly different values of tj,

t1 + ∆t1, t2 + ∆t2, . . . , tN + ∆tN

3. Solve a linear system for the differences

(∆t1, ∆t2, . . . , ∆tN)

to make the coefficients bk equal to zero (approximately).

Fornberg’s Method

COST OF ONE ITERATION: O(N log N)

RATE OF CONVERGENCE: linear

GENERALITY: requires close initial solution

9. Generalities on Conformal Mapping Methods

We have mentioned that there are a great many conformal mapping methods,each based on a specific property of functions of a complex variable. Here wehave seen only a few. There are other better ones; for example, Wegmann’smethod [25] based on the Riemann-Hilbert problem, has the same O(N log N)iteration cost and offers quadratic convergence. We suggest the following generalapproach for discovering new such methods.

• Choose a characterization of analytic mapping functions• Use it to measure in some sense how much a given function falls short of

this criterion• Write an equation to describe this numerically• Apply some numerical method to solve this equation• . . . and hope that it works!

10. Interpolating Polynomial Method

We describe here the method presented in [18], together with some of theideas that led to its development, and some ideas for variations of the method.We take as the first step in our heuristic procedure the following fact proved byWegmann [24].


e2πk

Ni -

PP (e

2πk

Ni)wk

Figure 22. Polynomial approximation of f

Theorem 5. For ∂D sufficiently smooth and for N sufficiently large, there is a

unique (suitably normalized) polynomial of degree N + 1 close to f which takes

the 2N -th roots of unity to points cyclicly ordered along ∂D. These polynomials

approach f as N → ∞.

Let us explain what is involved in this result. Recall that our problem is tofind wj ∈ ∂D such that e

2πij/N ↔ wj.

Consider some points w1, w2, . . . , wN along ∂D. Let us investigate the hypoth-

esis that these indeed could be the values of f(e2πk

Ni) for a Riemann mapping f

from the unit disk to D.

Let us suppose that f is to be approximated by a polynomial P (z) of degree

N . Now, it is very easy to find a polynomial P (z) such that P (ej 2πi

N ) = wj,j = 1, . . . , N . Thus the existence of such a polynomial tells us nothing aboutwhether (wj) are the right points or not for f .

For N fixed, let us write ζj = ej 2πi

N . Wegmann’s theorem says that if (wj)are the right points for f at ζk, then the polynomial approximation P will notonly be right for f at ζj, but also at ζj+ 1

2

. For ease of reference, I will call

ζj+ 1

2

the “half-click” points. In Figure 23 we take D = D, z0 = 0, so that the

solution is f = identity (or any rotation). We have deliberately taken wk not tobe equally spaced along ∂D, and then calculated the image of all of ∂D underthe corresponding interpolating polynomial. The images of the half-click pointsare marked with an ×.

When wj are not the correct values for f(ζj), then the polynomial approxima-tion may be very far from ∂D at the half-click points. Further, the image P (∂D)may not be a simple curve.

This leads us to phrase the criterion for the conformal mapping: first wediscretize the problem to the N points ζj. Our complex-analysis criterion is

• The truncated power series of f of degree N maps all the2N -th roots of unity close to ∂D.

The next step in the development of our algorithm is to “measure” how mucha given set of guessed points (wj) fails to fulfill this criterion.

We choose N to be a power of 2 (to facilitate the numerical work). Given~w = (wj), there is a unique P = P~w, the interpolating polynomial for ~w, such


that

(10) P~w is a polynomial of degree ≤ N ,

P~w(0) = 0,

P~w(e2πi

Nj) = wj for 0 ≤ j ≤ N − 1.

Since P is easy to calculate, so are its half-click values

(11) uj = P~w

(e

2πi

N(j+ 1

2)

).

The answer is given by a matrix multiplication,

(12) ~u = C ~w,

where

cj =−1

N+

i

Ncot(

π

N(j +

1

2)),(13)

Cjk = cj−k).

C is a circulant matrix, and C ~w can be calculated via Fast Fourier Transform(FFT) in O(N log N) operations. This is also a key feature of Fornberg’s method.

10.1. Some properties of the half-click mapping. C is an orthogonal ma-trix, and satisfies

C2 = E = shift left by one index,

C = R[1/2]

,

E = R[1]

whereR

[β](w) = w[β] = (P~w(eiβ

ej 2πi

N ))j=0,...,N−1.

By analogy one may call R a “β-click” of the W -values.

Thus we have uj = P~w(wj), ~u = C(~w), C(~u) = E(~w), (C(~u))j = uj−1.With this we can measure the discrepancy of a given ~w from the criterion ofWegmann’s theorem: define ρ = ρD to be the projection onto the nearest point

wk

uk

wk+1

Figure 23. Image of ∂D under interpolating polynomial


w∗j+1

u∗j

ρ(u∗j)

w∗j

?

ρ

HHYC

Figure 24. ~w 7→ ρ(C(~w))

of ∂D = γ([0, 2π]). This is defined for points sufficiently near to ∂D. Then thediscrepancy can be described by the N -vector

(uj − ρ(uj))j.

10.2. Interpolating polynomial algorithm. There are variations on this wayof describing the discrepancy. Begin with trial values

~w = (w0, w1, . . . , wN−1) ∈ (∂D)N.

Calculate~w 7→ ρ(C(~w)).

If ~w were the true solution, then ρ C(~w) would be the same as C(~w) ∈ ∂D.

If we repeat the process, then the image of wj should go over to wj+1, sinceC

2 = E. This suggests looking at ρ C ρ C and comparing it with the shiftE. We define the basic step of the algorithm as something very similar, and evenmore convenient:

(14) Φ = ΦD = ρ C−1 ρ C

When (~w) is a fixed point of Φ, it follows that ρ(uj) = uj, so uj ∈ ∂D as required.

The numerical method for solving Φ(~w) = ~w, as it was in Theodorsen’smethod, is simply by iteration towards an attractive fixed point. When the orig-inal ~w

∗ is close enough to ∂D, the convergence to a fixed point of Φ is generallylinear.

To see more clearly why “ρC” appears twice in the definition of Φ, note thatthe space of solutions normalized by f(0) = 0 can be identified with the circleS

1, being formed offβ(z) = f(eiβ

z), 0 ≤ β < 2π.

We can think of ~w = (w0, w1, . . . , wN−1) being shifted (or rotated) along ∂D

to ~w[β] = (w

[β]

0, w

[β]

1, . . . , w

[β]

N−1). The “ρC” algorithm approaches this space of

solutions, but not a particular solution.

The convergence characteristics of the Interpolating Polynomial method arethe same as for Fornberg’s. Recall that we have fixed N , and Φ depends uponN . Therefore a solution of Φ(~w) = ~w is not a true solution of the mappingproblem, but rather a solution to the truncated problem for N -th degree. A true


algorithm for approximating the Riemann mapping to arbitrary accuracy wouldhave to involve a step for increasing the value of N at appropriate moments.

10.3. Numerical example. For the first example, we will calculate the sameRiemann mapping as did Fornberg in 1980. His domain Dα, shown in Figure 25,is bounded by the curve

(15) ((Re w − .5)2 + (Im w − α)2)(1 − (Re w − .5)2 − (Im w)2) − 1 = 0

0.5 1.

-1.

0.5

1.2.0

0

1.0

0.5

Figure 25. Domain defined by equation (15) for α = 0.5, 1.0, 2.0

Iterating Φγ: Method of Fornberg:

α N FFTs ‖∆~w‖∞ ‖∆~a‖∞ FFTs Accuracy of

Taylor coefs.

2.0 4 16 .48×10−1

2.0 8 4 .89×10−1

2.0 16 4 .16×10−1

2.0 32 4 .18×10−2

2.0 64 8 .91×10−5

2.0 128 0 —

1.5 128 24 .13×10−8 .70×10−9 96 .14×10−7

1.2 128 36 .45×10−8 .37×10−9 100 .33×10−5

1.2 256 0 — 50 .97×10−8

1.0 256 48 .68×10−7 .63×10−8 108 .17×10−5

1.0 512 0 — 50 .40×10−8

.9 512 44 .53×10−7 .82×10−8 116 .26×10−6

.9 1024 0 — 89 .98×10−10

.8 1024 80 .35×10−8 .33×10−9 155 .42×10−7

.8 2048 0 — 81 .55×10−11

.75 2048 112 .41×10−9 .37×10−10 143 .58×10−9

.72 2048 108 .40×10−8 .35×10−9 151 .40×10−8

.7 2048 108 .95×10−8 .81×10−9 120 .30×10−7

.7 4096 0 — 81 .31×10−11

.68 4096 148 .12×10−9 .97×10−11 151 .38×10−10

.66 4096 148 .53×10−9 .42×10−10 151 .13×10−9

.64 4096 136 .92×10−8 .69×10−9 159 .29×10−8

.62 4096 136 .33×10−7 .23×10−8 140 .19×10−7

.6 4096 144 .48×10−7 .33×10−8 148 .84×10−7

.6 8192 0 — 93 .23×10−10

.58 8192 168 .19×10−7 .12×10−8 167 .78×10−9

.56 8192 188 .14×10−7 .83×10−9 171 .90×10−9

.54 8192 180 .11×10−6 .63×10−8 171 .22×10−7

.54 16384 0 — 77 .28×10−10

.52 16384 180 .46×10−6 .23×10−7 178 .99×10−9

.5 16384 212 .25×10−6 .11×10−7 228 .15×10−7

Figure 26. Table of comparative behaviour of Fornberg’s methodand the Interpolating Polynomial method


-0.5 0.5 1 1.5

-1

-0.5

0.5

1

-0.5 0.5 1 1.5

-1

-0.5

0.5

1

True values of ~w Initial test values ~w∗

Figure 27.

with a variable parameter α. The high crowding factor for the domain D0.5

mentioned earlier is largely due to the fact that the base point z0 = 0 is nearthe left-hand side of the domain. If one starts with evenly spaced points alongthe boundary, then probably none of the “fast” methods will converge to thesolution. In fact, one has to be extremely close to the solution for convergenceto be possible. One way to get around this situation is to solve first the mappingproblem for D2.0, which is nearly circular, and then project the solution pointsto say, D1.5, solve the problem there, and then project to another nearby Dα.This was done in the same way for the Interpolating Polynomial method in [18],and the fairly similar results make one wonder whether the two methods in somesense may be based on essentially the same fundamental ideas.

Now we look at a simpler domain, a unit disk which is centered at α, 0 < α < 1.The mapping function is

(16) f(z) = α − α − z

1 − αz.

Figure 28. P (∂D) for initial test values of Figure 27


Figure 29. Left: P (∂D) after one application of the Interpolat-ing Polynomial algorithm. Center: Result after second application.Right: Imperfect view of domain due to discretization

For illustration we take α = 0.5 and N = 32. Supposing that we don’t knowthe formula (16), we will naıvely guess that the wj are equally spaced; i.e., wetake wj = ζj as on the right of Figure 27. The resulting P (∂D) turns out to bethe rather complicated curve shown in Figure 28, a very bad approximation ofthe circumference of the disk.

After one iteration of Φ, a most of the wj have moved over to the left closer towhere they belong, and the image curve looks a bit more like ∂D. At the seconditeration they cannot be distinguished visually from the true positions, and thehalf-click images uj appear to lie exactly on ∂D as well.

Of course, this is only an approximation, and to study its accuracy one maygraph the change in ~w∗ from one iteration to the next. For this any convenientnorm will do; in Figure 30 we use log || ||∞.

In Figure 31 we show the results for 0, 1, and 2 iterations of the methodfor an ellipse and a square, starting in each case with equally spaced boundarypoints. Note that a polynomial of degree 32 does not give a particularly goodapproximation of the Riemann mapping for a square.

-3

-2.5

-2

-1.5

-1

-0.5

0

Figure 30. Logarithmic graph of amount ~w is moved in each iteration


Figure 31. Interpolating polynomial algorithm applied to ellip-tical and square domains.

11. The Quest for Better Methods

Once we have the basic idea of the solution of the mapping problem being afixed point for Φ, we can look for ways to calculate it more rapidly. Here we willdescribe a few attempts.

11.1. Methods using derivatives. Given ~w on ∂D, we can write the interpo-lating polynomial P = P~w explicitly, and thus can calculate its derivative, givingan N -vector

(17) w′j = P

′~w(ej 2πi

N ) = (C ′~w)j

for an appropriate matrix C′.

On the other hand, there are many other formulas involving f′. For example,

from (2) we have ieiθf′(eiθ) = γ

′(b(θ))b′(θ) so we can write

(18) ~b′ = i ~ζ

1

~γ′ ~w′.

Many ideas present themselves for combining (17) and (18). For example, given~b one calculates ~w and ~γ

′, and then can obtain ~b′ from which a new value of ~b can

be estimated. Alternatively, from an initial ~b written as a deviation ~b = ~b∗ + ∆~b

from the true solution ~b∗, to obtain an equation for ∆~w. However, so far I have

not been able to create an algorithm which converges by using any such idea.

11.2. Method of simultaneous interpolation. Let ~w∗ ⊆ (∂D)N denote the

true solution of the mapping problem. Let ~w ⊆ (∂D)N be an approximation. Wewould like to devise an algorithm which moves ~w closer to ~w

∗; or in other words,to calculate the difference ∆~w approximately, and then add it to ~w to find ~w

∗.Using the parameters ~t = (tj) ∈ R

N , we see that within a neighborhood of each


γ(tj) = wj the curve γ = ∂D can be approximated by a series, which we writesymbolically as

(19) γ(~t + ∆~t) = ~w + ~γ′∆~t +

~γ′′

2∆~t

2 + . . .

In this formula we use coordinate-wise multiplication in a natural way, whichmeans a subscript “j” can be applied to all the letters. In particular, we wantto find the value of ∆~t for which (19) is equal to ~w

∗.

To find it, write ~u = C ~w. Then near ρ~u = γ(~s), ∂D is approximated by asimilar series

(20) γ(~s + ∆~s) = ~w + ~γ′∆ρ~u + ~τ

′∆~s +~τ′′

2~∆s

2 + . . . .

By the same token, there should be a value of ∆~s for which (20) is equal to~u∗ = ρ~u

∗.

First we will examine the linear approximations obtained by truncating (19),(20):

~w∗ = ~w + ~γ

′∆~t , ~u∗ = ρ~u + ~τ

′∆~s

which are connected by the relation

C(~w + ~γ′∆~t ) = ρ~u + ~τ

′∆~s.

One solves this to find that

(21) ImC(~γ′∆~t )j

τ′j

= Imρuj − uj

τ′j

since ∆sj ∈ R. The real-linear operator

∆~t 7→ ImC(~γ′∆~t)

τ ′

from Rn to R

n can be seen to have a null vector ~d close to ~1 = (1, 1, . . . ). It

can be found by standard conjugate gradient methods, and ∆~t = ~d gives anapproximate solution of the mapping problem since ρ~u = ~u by (21).

Now we will use a quadratic approximation. The relation is

C( ~w + ~γ′∆~t +

~γ′′

2∆~t

2

) = ρ~u + ~τ′∆~s +

~τ′′

2∆~s

2.

From what we already know concerning the linear approximation, we maycancel several terms when this is expanded. We find thus a quadratic relationbetween ∆~t and ∆~s. The linear system already gave an approximation for ∆~t,which we substitute to solve for ∆~s.

The linear and quadratic versions of “simultaneous interpolation” method givefairly good convergence, as shown in Figure 32. There are two disappointingfacts. One is that we have obtained slightly better convergence at a much highercalculation cost. The other is that, according to these experiments, the quadraticmethod seems no better than the linear one, in spite of costing a good deal morework for each iteration.


-15

-12.5

-10

-7.5

-5

-2.5

0

Figure 32.

11.3. Minimization approach. We are looking for u∗j ∈ ∂D, so we try to

minimize the quantity||(I − ρ)C ~w||2

2

for ~w ∈ (∂D)N . Here || · ||2 refers to the L2 norm. We calculate the Jacobian ofthe projection map at v near u. Because of the approximate relation

ρv = ρu + (u − ρu) i Imt − ρu

u − ρu,

we find that the Jacobian mapping is

Jρ(v) =1

|a|

((Re a)2 −Re a Im a

−Re a Im a (Im a)2

)

where a = u−ρu. Then the gradient of the real-valued function can be calculatedto be

||ϕ(~t)||22

= ||(I − ρ)C(~w∗ + ~γ

′~t)||22

which is given by

∇||ϕ||22

= 2JTϕ ϕ

= 2 JT~γ′ C

tJ

TI−ρ ϕ.

Once one has the gradient, one can use it to find a value of ∆~t which minimizes thefunction. Some experimentation has shown that this approach works, althoughso far not very well.

12. Combined Methods

Recall that the “easy” methods are of general application but converge slowly,and the “fast” methods only apply when one already has a good idea of wherethe solution is. Thus in practice it is only logical to combine the two approaches.

Given a domain D, the first step is to map it to the interior of D, and thenapply an osculation method to obtain an image domain which is nearly circular.Then one of the faster methods can be applied to this image domain.

Figure 33, taken from [19], explains this procedure, which can be automatedreasonably well. The initial domain is defined by several thousand points. The


1

i(a) (b) ( )

Figure 33. Combined method. (a) Original domain. (b)Nearly circular domain obtained by two iterations of an osculationmethod. (c) After applying the Interpolation Polynomial methodto (b), the inverse of the osculation result is applied. Note theeffect of the crowding phenomenon and discretization.

image under the osculation mapping has the same number M of points (herearound 8000). The “fast” method is applied with a relatively small number N

of points (here 512). This is done because the cost O(N log N) increases fairlyrapidly with N . Thus when the inverse of the osculation mapping is applied, weonly have N points to describe the domain. When the crowding phenomenonis present, this may cause part of the figure to be badly represented. On theother hand, if one is only interested in approximating the conformal mappingnear another part of the boundary (or in the interior), this may be a very usefulaspect of the method.

13. Epilogue

We suggest that the reader interested in knowing more about numerical con-formal mapping consult the following.

The books [23] and [14] give detailed explanations of a great number of map-pings with specific formulas. We mention also [3], a much older book, whichgives a general introduction to the theory of functions of a complex variable asnecessary to understand the topic of conformal mapping, as do [13], [16].

Reference [9], a half-century old text in German, is divided into two parts,covering precisely what we have called the “easy” and “fast” methods for confor-mal mapping. Bear in mind that the numerical examples were calculated withoutcomputers!

Reference [12] gives a much more modern and very practical treatment, in-cluding some of the methods we have described here (Koebe and Grassmannosculation, Theodorsen’s method). In [15] one may find a great variety of otherconformal mapping methods. As to detailed treatiseson specific methods, we rec-ommend the book [6] on the Schwarz-Christoffel method, and [20] which explainsthe method of circle packings.


Finally we recommend the survey article [4], which gives a more recent per-spective of the existing methods.

References

1. W. Abikoff, The uniformization theorem. Amer. Math. Monthly 88 (1981), 574–592.2. L. Ahlfors, Complex Analysis: An introduction to the theory of analytic functions of one

complex variable, Third edition, International Series in Pure and Applied Mathematics,McGraw-Hill Book Co., New York 1978.

3. L. Bieberbach, Conformal mapping, Chelsea, New York 1964.4. T. K. DeLillo, The accuracy of numerical conformal mapping methods: a survey of exam-

ples and results, SIAM J. Numer. Anal. 31 (1994) 788–812.5. John B. Conway, Functions of one complex variable, Second edition, Graduate Texts in

Mathematics 11, Springer-Verlag, New York-Berlin 1978.6. T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel mapping, Cambridge Monographs

on Applied and Computational Mathematics, Cambridge University Press, Cambridge2002.

7. T. A. Driscoll and S. A. Vavasis, Numerical conformal mapping using cross-ratios andDelaunay triangulation, SIAM J. Sci. Comput. 19 (1998) 1783-1803.

8. B. A. Fornberg, A numerical method for conformal mapping of doubly connected regions,SIAM J. Sci. Statist. Comput. 5 (1984) 771–783.

9. D. Gaier, Konstruktive Methoden der konformen Abbildung, Springer tracts in naturalphilosophy, v. 3, Springer, Berlin 1964.

10. E. Grassmann, Numerical experiments with a method of successive approximation forconformal mapping. Z. Angew. Math. Phys. 30 (1979) 873–884.

11. J. Heinhold, R. Albrecht, Zur Praxis der konformen Abbildung, Rend. Circ. Mat. Palermo3 (1954) 130–148.

12. P. Henrici, Applied and computational complex analysis, Vol. 3, Pure and Applied Mathe-matics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York1986.

13. E. Hille, Analytic function theory, Chelsea Publishing Company, New York 1959.14. H. Kober, Dictionary of conformal representations, Dover, New York 1957.15. P. K. Kythe, Computational conformal mapping, Boston: Birkhuser, Boston 1998.16. Z. Nehari, Conformal mapping, McGraw-Hill, New York 1952.17. S. T. O’Donnell and V. Rokhlin, A fast algorithm for the numerical evaluation of conformal

mappings, SIAM J. Sci. Statist. Comput. 10 (1989) 475–487.18. R. M. Porter, An interpolating polynomial method for numerical conformal mapping. SIAM

J. Sci. Comput. 23 (2001) 1027–1041.19. R. M. Porter, An accelerated osculation method and its application to numerical conformal

mapping, Complex Var. Theory Appl., 48 (2003) 569–582.20. K. Stephenson, Introduction to circle packing: The theory of discrete analytic functions,

Cambridge University Press, Cambridge, 2005.21. L. N. Trefethen and T. A. Driscoll, A. Schwarz-Christoffel mapping in the computer era.

Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), Doc.Math. 1998, Extra Vol. III, 533–542.

22. M. R. Trummer, An efficient implementation of a conformal mapping method based onthe szego kernel, SIAM J. Numer. Anal. 23 (1986) 853–872.

23. W. von Koppenfels, Praxis der konformen Abbildung, Springer-Verlag, Berlin-Gottingen-Heidelberg 1959.

24. R. Wegmann, Discrete Riemann-Hilbert problems, interpolation of simply closed curves,and numerical conformal mapping, J. Comput. Appl. Math. 23 (1988) 323–352.


25. R. Wegmann, Conformal mapping by the method of alternating projections, Numer. Math.56 (1989) 291–307.

R. Michael Porter E-mail: [email protected]: Department of Mathematics, Centro de Investigacion y de Estudios Avanzados del

I.P.N., Apdo. Postal 14-740, 07000 Mexico, D.F., Mexico


Introduction to quasiconformal mappings in n-space

Antti Rasila

Abstract. We give an introduction to quasiconformal mappings in the Eu-clidean space R

n.

Keywords. quasiconformal mappings.

2000 MSC. 30C65.

Contents

1. Introduction: Mercator’s map 240

2. History and background 241

3. Preliminaries 242

ACLp functions 242

Conformal mappings 242

Mobius transformations 242

4. Modulus of a path family 244

Ring domains 248

Modulus in conformal mappings 250

Capacity of a condenser 251

Sets of zero capacity 252

Spherical symmetrizations 252

Canonical ring domains 253

Spherical metric 256

5. Quasiconformal mappings 256

Examples 257

6. An application of the modulus technique 258

References 259

Version October 19, 2006..

240 A. Rasila IWQCMA05

1. Introduction: Mercator’s map

Perhaps the greatest cartographer of the time, Gerardus Mercator (5 March1512 – 2 Dec 1594) was born Gerhard Kremer of German parents in the townof Rupelmonde near Antwerp. Like many other intellectuals of his time, heLatinized his German name, which meant “merchant”, and changed it to thename Mercator which means “world trader”. Mercator was a mapmaker, scholar,and religious thinker. His interests ranged from mathematics to calligraphy andthe origin of the universe. Mercator studied mathematics in Louvain under thesupervision of mathematician and astronomer Gemma Frisius.

Figure 1: Gerardus Mercator (source: Wikipedia) and a World map using theMercator projection.

The Mercator map is defined by the formula

(x, y) =(λ, log

(tan(π/4 + φ/2)

)),

where φ is the latitude and λ is the longitude of the point on the sphere. Mercatorpublished the first map using this projection in 1569, a wall map of the worldon 18 separate sheets entitled: “New and more complete representation of theterrestrial globe properly adapted for its use in navigation.” The projection didnot become popular until 30 years later (1599), when Edward Wright publishedan explanation of it. An important property of the Mercator projection is thatit is conformal, i.e. the angles are preserved.

Figure 2: India and Finland in the Mercator projection.

Introduction to quasiconformal mappings in n-space 241

The Mercator projection is not without flaws, however. For example, fromthe picture above one might conclude that India is approximately twice as largeas Finland. Actually, India’s land area is 3, 287, 590 km2, almost ten times thatof Finland (338, 145 km2). This example also illustrates the reasons why we aremainly interested in the local distortion of the geometry in this theory.

2. History and background

Conformal mappings play extremely important role in complex analysis, aswell as in many areas of physics and engineering. The class of conformal map-pings turned out to be too restrictive for some problems. Quasiconformal map-pings were introduced by H. Grotzsch provide more flexibility in 1928. Importantresults were also obtained by O. Teichmuller and L. V. Ahlfors [1]. A compre-hensive survey on quasiconformal mappings of the complex plane is [16]. Seealso [15].

By the Riemann mapping theorem a simply-connected plane domain withmore than one boundary point can be mapped conformally onto the unit disk B2.On the other hand, Liouville’s theorem says that the only conformal mappings inR

n, n ≥ 3, are the Mobius transformations. Hence the plane theory of conformalmappings does not directly generalize to the higher dimensions.

Quasiconformal maps were first introduced in higher dimensions by M. A. Lav-rent’ev in 1938. The systematic study of quasiconformal maps in R

n was begunby F. W. Gehring [5] and J. Vaisala [20] in 1961. Since then the theory andit’s generalizations have been actively studied [3, 4, 21, 23]. Generalizationsinclude quasiregular [18, 22, 19] and quasisymmetric mappings, and recently themappings of finite distortion [13] and the quasiconformal mappings in the metricspaces [10, 11, 12].

Quasiconformal mappings in Rn are natural generalization of conformal func-

tions of one complex variable. Quasiconformal mappings are characterized bythe property that there exists a constant C ≥ 1 such that the infinitesimallysmall spheres are mapped onto infinitesimally small ellipsoids with the ratio ofthe larger “semiaxis” to the smaller one bounded from above by C.

l

L

Figure 3: Image of a small sphere.


For a comprehensive historical review of the theory of quasiconformal map-pings in both plane and space settings, see [2]. A survey of the theory of qua-siconformal mappings is given in [8] (see also [14]). This presentation is for themost parts based on [7], [21] and [22].

3. Preliminaries

We shall follow standard notation and terminology adopted from [21], [22]and [19]. For x ∈ R

n, n ≥ 2, and r > 0 let Bn(x, r) = z ∈ Rn : |z − x| < r,

Sn−1(x, r) = ∂Bn(x, r), Bn(r) = Bn(0, r), S

n−1(r) = ∂Bn(r), Bn = Bn(1),Hn = x ∈ R

n : xn > 0, Bn+

= Bn ∩ Hn, and Sn−1 = ∂Bn. For t ∈ R

and a ∈ Rn \ 0, P (a, t) = x ∈ R

n : x · a = t ∪ ∞, is a hyperplane inR

n= R

n ∪ ∞ perpendicular to the vector a and at distance t/|a| from theorigin. The surface area of S

n−1 is denoted by ωn−1 and Ωn is the volume of Bn.It is well known that ωn−1 = nΩn and that

Ωn =π

n/2

Γ(1 + n/2)

for n = 2, 3, . . ., where Γ is Euler’s gamma function. The standard coordinateunit vectors are denoted by e1, . . . , en. The k-dimensional Lebesgue measureis denoted by mk. For k = n we omit the subscript and denote the Lebesguemeasure on R

n simply by m.

For nonempty subsets A and B of Rn, we let d(A) = sup|x − y| : x, y ∈ A

be the diameter of A, d(A,B) = inf|x−y| : x ∈ A, y ∈ B the distance betweenthe sets A and B, and in particular d(x,B) = d(x, B).

ACLp functions. Let Q be a closed n-interval x ∈ Rn : ai ≤ xi ≤ bi, i =

1, . . . , n. A function f : Q → Rm is called ACL (absolutely continuous on lines)

if f is continuous and if f is absolutely continuous on almost every line segment inQ parallel to one of the coordinate axes. Let U be an open set in R

n. A functionf : U → R

m is ACL if f |Q is ACL for every closed n-interval Q ⊂ U . Such afunction has partial derivatives Dif(x) a.e. in U , and they are Borel functions[21, 26.4]. If p ≥ 1 and the partial derivatives of f are locally L

p-integrable, f issaid to be in ACLp or in ACLp(U).

Conformal mappings. Let G,G′ be domains in R

n. A homeomorphism f : G →G

′ is called conformal if f is in C1(G), Jf (x) 6= 0 for all x ∈ G, and |f ′(x)h| =

|f ′(x)||h| for all x ∈ G and h ∈ Rn. If G,G

′ are domains in Rn, a homeomorphism

f : G → G′ is conformal if its restriction to G \ ∞, f

−1(∞) is conformal.

Mobius transformations. A Mobius transformation is a mapping f : Rn → R

n

that is composed of a finite number of the following elementary transformations:

(1) Translation: f1(x) = x + a.(2) Stretching: f2(x) = rx, r > 0.(3) Rotation: f3 is linear and |f3(x)| = |x| for all x ∈ R

n.


(4) Reflection in plane P (a, t):

f4(x) = x − 2(x · a − t)a

|a|2 , f4(∞) = ∞.

(5) Inversion in a sphere Sn−1(a, r):

f5(x) = a +r2(x − a)

|x − a|2 , f5(a) = ∞, f5(∞) = a.

In fact every Mobius transformation can be expressed as a composition of a finitenumber of reflections and inversions. It is easy to see that every elementarytransformation, and hence every Mobius transformation, is conformal.

Let a, b, c, d be distinct points in Rn. We define the absolute (cross) ratio by

(3.1) |a, b, c, d| =|a − c| |b − d||a − b| |c − d| .

This definition can be extended for a, b, c, d ∈ Rn

by taking limit.

An important property of Mobius transformations is that they preserve theabsolute ratios, i.e.

|f(a), f(b), f(c), f(d)| = |a, b, c, d|,

if f : Rn → R

nis a Mobius transformation. In fact, a mapping f : R

n → Rn

is aMobius transformation if and only if f preserves all absolute ratios.

Let a∗ = a/|a|2 for a ∈ R

n \ 0, 0∗ = ∞ and ∞∗ = 0. Fix a ∈ Bn \ 0. Let

σa(x) = a∗ + r

2(x − a∗)∗, r

2 = |a|2 − 1

be an inversion in the sphere Sn−1(a∗

, r) orthogonal to Sn−1. Then σa(a) = 0,

σa(a∗) = ∞. Let pa denote the reflection in the (n−1)-dimensional plane P (a, 0)

through the origin and orthogonal to a, and define a sense preserving Mobiustransformation by Ta = pa σa. Then Ta(B

n) = Bn and Ta(a) = 0. For a = 0we set Ta = id, i.e. the identity map.

0 a

r1S

a

S (a ,r)

n−1

*

*n−1

Figure 4: Construction of the Mobius transformation Ta.


4. Modulus of a path family

A path in Rn is a continuous mapping γ : ∆ → R

n, where ∆ is a (possiblyunbounded) interval in R. The path γ is called closed or open according as ∆ iscompact or open. The locus |γ| of γ is the image set γ∆.

Let γ : [a, b] → Rn be a closed path. The length ℓ(γ) of the path γ is defined

by means of polygonal approximation (see [21], pages 1-8). The path γ is calledrectifiable if ℓ(γ) < ∞ and locally rectifiable if each closed subpath of γ isrectifiable. If γ is a rectifiable path, then γ has a parameterization by means of arclength, also called the normal representation of γ. The normal representation ofγ is denoted by γ

0 : [0, ℓ(γ)] → Rn. By making use of the normal representation,

one may define the integral over a locally rectifiable path γ.

Definition 4.1. Let Γ be a path family in Rn, n ≥ 2. Let F(Γ) be the set of all

Borel functions ρ : Rn → [0,∞] such that

∫

γ

ρ ds ≥ 1

for every locally rectifiable path γ ∈ Γ. The functions in F(Γ) are called admis-

sible for Γ. For 1 < p < ∞ we define

(4.2) Mp(Γ) = infρ∈F(Γ)

∫

Rn

ρpdm

and call Mp(Γ) the p-modulus of Γ. If F(Γ) = ∅, which is true only if Γ containsconstant paths, we set Mp(Γ) = ∞. The n-modulus or conformal modulus isdenoted by M(Γ).

Lemma 4.3. [21, 6.2] The p-modulus is an outer measure in the space of all path

families in Rn. That is,

(1) Mp(∅) = 0,

(2) If Γ1 ⊂ Γ2 then Mp(Γ1) ≤ Mp(Γ2), and

(3) Mp

( ⋃j Γj

)≤ ∑

j Mp

(Γj

).

Proof. (1) Since the zero function is admissible for ∅, Mp(∅) = 0.

(2) If Γ1 ⊂ Γ2 then F(Γ2) ⊂ F(Γ1) and hence Mp(Γ1) ≤ Mp(Γ2).

(3) We may assume that Mp(Γj) < ∞ for all j. Let ε > 0. Then we canchoose for each j a function ρj admissible for Γj such that

∫

Rn

ρpj dm ≤ Mp(Γj) + 2−j

ε.

Now letρ = sup

j

ρj, Γ =⋃

j

Γj.

Then ρ : Rn → [0,∞] is a Borel function. Moreover, if γ ∈ Γ is locally rectifiable,

then γ ∈ Γj for some j, ∫

γ

ρ ds ≥∫

γ

ρj ds ≥ 1,


and hence ρ is admissible for Γ. Now

Mp(Γ) ≤∫

Rn

ρpdm ≤

∫

Rn

∑

j

ρpj dm ≤

∑

j

Mp(Γj) + ε.

By letting ε → 0, the claim follows.

Let Γ1 and Γ2 be path families in Rn. We say that Γ2 is minorized by Γ1 and

write Γ1 < Γ2 if every γ ∈ Γ2 has a subpath in Γ1.

Lemma 4.4. If Γ1 < Γ2 then Mp(Γ1) ≥ Mp(Γ2).

Proof. If Γ1 < Γ2 then obviously F(Γ1) ⊂ F(Γ2). Hence Mp(Γ1) ≥ Mp(Γ2).

Lemma 4.5. Let G be a Borel set in Rn, r > 0 and let Γ be the family of paths

in G such that ℓ(γ) ≥ r. Then Mp(Γ) ≤ m(G)r−p.

Proof. The claim follows immediately from (4.2) and the fact that the functionρ = χG/r is admissible for Γ.

Lemma 4.6. Path family Γ has zero p-modulus if and only if there is an admis-

sible function ρ ∈ F(Γ) such that∫

Rn

ρpdm < ∞ and

∫

γ

ρ ds = ∞

for every locally rectifiable path γ ∈ Γ.

Proof. If ρ satisfies the above conditions, clearly ρ/k is admissible for Γ for allk = 1, 2, . . . . Hence

Mp(Γ) ≤ k−p

∫

Rn

ρpdm → 0

as k → ∞, and thus Mp(Γ) = 0.

Now let Mp(Γ) = 0 and choose a sequence of functions ρk ∈ F(Γ) such that∫

Rn

ρpk dm < 4−k

, k = 1, 2, . . . .

Define

ρ(x) =( ∞∑

k=1

2kρ

pk(x)

)1/p

,

and note that ∫

Rn

ρpdm < ∞.

On the other hand, ∫

γ

ρ ds ≥∫

γ

2k/pρk ds ≥ 2k/p → ∞

as k → ∞ for every locally rectifiable path γ ∈ Γ.

Corollary 4.7. Let Γ be a path family in Rn

and denote by Γr the family of all

rectifiable paths in Γ. Then M(Γ) = M(Γr).


The path families Γ1, Γ2, . . . are called separate if there exist disjoint Borelsets Ei such that

(4.8)

∫

γ

χRn\Eids = 0

for all locally rectifiable γ ∈ Γi, i = 1, 2, . . ..

Lemma 4.9. [19, Proposition II.1.5] Let Γ, Γ1, Γ2, . . . be a sequence of path fam-

ilies in Rn. Then

(1) If Γ1, Γ2, . . . are separate and Γ < Γj for all j = 1, 2, . . . , then

Mp(Γ) ≥∑

j

Mp(Γj).

Equality holds if Γ =⋃

j Γj.

(2) If Γ1, Γ2, . . . are separate and Γj < Γ for all j = 1, 2, . . . , then

Mp(Γ)1/(1−p) ≥∑

j

Mp(Γj)1/(1−p)

, p > 1.

Proof. (1) Let ρ be admissible for Γ, and let Ej be as in (4.8). Then for allindices j the function ρj = χEj

ρ is admissible for Γj. It follows that

∑

p

Mp(Γj) ≤∑

j

∫

Rn

ρpj dm =

∑

j

∫

Ej

ρpdm ≤

∫

Rn

ρpdm.

(2) Let Ej be as in (4.8), and let E =⋃

j Ej. Then for all indices j the

function χEjρ is admissible for Γj. Let (aj) be a sequence such that aj ∈ [0, 1]

and∑

j aj = 1. Let

ρ =∞∑

j=1

ajχEjρj.

Next we show that ρ is admissible for Γ. Fix a locally rectifiable path γ ∈ Γ anda subpath γj ∈ Γj for each j = 1, 2, . . . . Now

∫

γ

ρ ds =

∫

γ

( ∑

j

ajχEjρj

)ds =

∑

j

aj

∫

γ

χEjρj ds

≥∑

j

aj

∫

γj

χEjρj ds ≥

∑

j

aj = 1.

Hence ρ is admissible for Γ and

Mp(Γ) ≤∫

Rn

ρpdm =

∫

E

ρpdm

=∑

j

∫

Ej

(∑

k

akχEkρ

)p

dm =∑

j

∫

Ej

apjρ

pj dm

≤∫

Rn

∑

j

apjρ

pj dm ≤

∑

j

apj

∫

Rn

ρpj dm.


By taking the infimum over all admissible ρj, we obtain

(4.10) Mp(Γ) ≤∑

j

apjMp(Γj).

We may assume that Mp(Γ) > 0 (if that would not be the case, the left side ofthe inequality is ∞ and there is nothing to prove). Hence by Lemma 4.4 we haveMp(Γj) ≥ Mp(Γ) > 0. Similarly, we may assume that Mp(Γj) < ∞.

Let

tk =1

∑k

j=1Mp(Γj)1/(1−p)

, aj,k = Mp(Γj)1/(1−p)

tk,

for j = 1, . . . , k and k = 1, 2, . . . . Now∑k

j=1aj,k = 1. We choose aj,k = 0 for

j ≥ k + 1, and by (4.10) we have

Mp(Γ) ≤ tpk

k∑

j=1

Mp(Γj)p/(1−p)Mp(Γj) =

( k∑

j=1

Mp(Γj)1/(1−p)

)1−p

.

By letting k → ∞ the claim follows.

For E,F,G ⊂ Rn we denote by ∆(E,F ; G) the family of all nonconstant paths

joining E and F in G.

Lemma 4.11. [22, 5.22] Let p > 1 and let E,F be subsets of Hn. Then

Mp(∆(E,F ;Hn)) ≥ 1

2Mp(∆(E,F )).

E

F

G h

Figure 5: Cylinder with bases E and F .

Example 4.12. Let E ⊂ x ∈ Rn : xn = 0 be a Borel set, h > 0, F = E +hen.

We define a cylinder G with bases E,F by

G = x ∈ Rn : (x1, . . . , xn−1, 0) ∈ E, 0 < xn < h.

Then Mp(∆(E,F ; G)) = mn−1(E)h1−p = m(G)h−p.


Proof. Choose ρ ∈ F(Γ) where Γ = ∆(E,F ; G)) and let γy be the verticalsegment from y ∈ E. Then γy ∈ Γ. We note that 1/p+(p− 1)/p = 1, and henceby Holder’s inequality

1 ≤( ∫

γy

ρ ds

)p

≤( ∫

γy

1 ds

)p−1( ∫

γy

ρpds

)= h

p−1

∫

γy

ρpds.

This holds for all y ∈ E and hence by the Fubini theorem∫

Rn

ρpdm ≥

∫

E

( ∫

γy

ρpds

)dmn−1 ≥

mn−1(E)

hp−1.

Since the above holds for any ρ ∈ F(Γ),

Mp(Γ) ≥ mn−1(E)

hp−1.

Next we choose ρ = 1/h inside G and ρ = 0 otherwise. Then ρ is admissible forΓ and

Mp(Γ) ≤∫

Rn

ρpdm =

mn−1(E)

hp−1.

Remark 4.13. In Example 4.12 the modulus is invariant under similarity map-pings if and only if p = n. This is the reason why the case p = n is so importantin the theory of quasiconformal mappings. Later in this section we will showthat M(Γ) is a conformal invariant.

Ring domains. A domain G in Rn

is called a ring, if Rn \ G has exactly two

components. If the components are E and F , we denote the ring by R(E,F ).

In general, it is difficult to calculate the modulus of a given path family. Nexttwo lemmas give us an important tool, letting us to obtain effective upper andlower bounds for the modulus in many situations.

a

b

Figure 6: Spherical ring with 0 < a < b < ∞.


Lemma 4.14. [21, 7.5] Let 0 < a < b < ∞, A = Bn(b) \ Bn(a) and

ΓA = ∆(S

n−1(a), Sn−1(b); A).

Then

M(ΓA) = ωn−1

(log

b

a

)1−n

.

Proof. Let ρ ∈ F(ΓA). For each unit vector y ∈ Sn−1 let γy : [a, b] → R

n theradial line segment defined by γy(s) = sy. As in Example 4.12 by Holder’sinequality we obtain

1 ≤( ∫

γy

ρ ds

)n

≤( ∫ b

a

ρ(sy)ns

n−1ds

)( ∫ b

a

1

sds

)n−1

=(

logb

a

)n−1∫ b

a

ρ(sy)ns

n−1ds.

By integrating over y ∈ Sn−1, we have

(4.15) ωn−1 ≤(

logb

a

)n−1∫

Rn

ρndm.

Taking the infimum over all admissible ρ yields

ωn−1 ≤(

logb

a

)n−1

M(ΓA).

Next we define ρ(x) = 1/(|x| log(b/a)

)for x ∈ A, and ρ(x) = 0 otherwise.

Clearly ρ is admissible for ΓA, and hence

M(ΓA) ≤∫

Rn

ρndm = ωn−1

(log

b

a

)−n∫ b

a

1

sds = ωn−1

(log

b

a

)1−n

.

Lemma 4.16. [21, 7.8] Let x0 ∈ Rn

and let Γ be the family of all nonconstant

paths through x0. Then M(Γ) = 0.

Proof. If x0 = ∞, the claim follows immediately from Corollary 4.7.

If x0 6= ∞, we let

Γk = γ ∈ Γ : |γ| ∩ Sn−1(x0, 1/k) 6= ∅.

We may assume that x0 = 0. Then for all R > 1/k

Γk > ∆R, where ∆R = ∆(S

n−1(1/k), Sn−1(R);Bn(R) \ Bn(1/k)

),

and by Lemma 4.4 and Lemma 4.14 we have

M(Γk) ≤ M(∆R) = ωn−1

(log

R

1/k

)1−n

→ 0

as R → ∞, and thus M(Γk) = 0. On the other hand, because Γ =⋃

k Γk we haveby Lemma 4.3 (3)

M(Γ) ≤∑

k

M(Γk) = 0.


Modulus in conformal mappings. Let G ⊂ Rn

and f : G → Rn

be a continu-ous function. Suppose that Γ is a family of paths in G. Then Γ′ = f γ : γ ∈ Γis a family of paths in f(G). Γ′ is called the image of Γ under f .

Theorem 4.17. [21, 8.1] If f : G → f(G) is conformal, then M(f(Γ)) = M(Γ)for all path families Γ in G.

Proof. By Lemma 4.16 we may assume that the paths of Γ, f(Γ) do not gothrough ∞. Let ρ1 ∈ F(f(Γ)), and define

ρ(x) = ρ1

(f(x)

)|f ′(x)|

for x ∈ G and ρ(x) = 0 otherwise. Because f is a conformal mapping (see [21,5.6]), ∫

γ

ρ ds =

∫

γ

ρ1

(f(x)

)|f ′(x)| |dx| =

∫

fγρ1 ds ≥ 1

for every locally rectifiable γ ∈ Γ. It follows that ρ ∈ F(Γ), and

M(Γ) ≤∫

Rn

ρndm =

∫

G

ρn1

(f(x)

)|Jf (x)| dm =

∫

f(G)

ρn1dm =

∫

Rn

ρn1dm

for all ρ1 ∈ F(f(Γ)), and thus M(Γ) ≤ M(f(Γ)). The inverse inequality followsfrom the fact that f

−1 is conformal.

Lemma 4.18. Let A ⊂ Hn, B ⊂ (∁Hn), Γ = ∆(A,B), and let

Γ1 = ∆(A, ∂Hn), Γ2 = ∆(B, ∂Hn).

Then

M(Γ) ≤ 2−n(M(Γ1) + M(Γ2)

).

In particular, the equality holds if A = g(B), where g is the reflection in Hn.

Proof. Let ρ1 ∈ F(Γ1) and ρ2 ∈ F(Γ2). We note that if γ ∈ Γ is a rectifiablepath, then γ has subpaths γ1, γ2 such that γ1 ∈ Γ1, γ2 ∈ Γ2. Thus

1 ≤ 1

2

∫

γ1

ρ1 ds +1

2

∫

γ2

ρ2 ds.

We define ρ = ρ1/2 + ρ2/2. Now ρ is an admissible function for the curve familyΓ and hence

M(Γ) ≤∫

Rn

ρndm.

We may assume that ρ1(z) = 0 for z /∈ Hn, and ρ2(z) = 0 for z ∈ Hn. Asρ = ρ1/2 + ρ2/2, we obtain

∫

Rn

ρndm = 2−n

∫

Hn

ρn1dm + 2−n

∫

Rn\Hn

ρn2dm

= 2−n( ∫

Rn

ρn1dm +

∫

Rn

ρn2dm

).


It follows that

M(Γ) ≤ 2−n(M(Γ1) + M(Γ2)

).

Next we consider the case A = g(B). Let ρ be an admissible function for thepath family Γ and denote ρ g by ρ. Now the function

ρ =

ρ + ρ on Hn

,

0 on ∁Hn,

is admissible for the path family Γ1. By the inequality (a + b)n ≤ 2n−1(an + bn)

(for a, b ≥ 0) and the fact that M(Γ1) = M(Γ2) it follows that

M(Γ1) ≤∫

Rn

ρndm =

1

2

∫

Rn

(ρ + ρ)ndm

≤ 2n−2

∫

Rn

(ρn + ρn)dm = 2n−1

∫

Rn

ρndm.

Hence,

M(Γ1) + M(Γ2) = 2M(Γ1) ≤ 2n

∫

Rn

ρndm,

for any ρ admissible for the curve family Γ. By taking infimum over all admissibleρ, the claim follows.

Capacity of a condenser. A condenser in Rn is a pair E = (A,C), where A

is open in Rn and C is a compact subset of A. The p-capacity of E is defined by

(4.19) cappE = infu

∫

A

|∇u|pdm, 1 ≤ p < ∞,

where the infimum is taken over all nonnegative functions u in ACLp(A) withcompact support in A and u|C ≥ 1. The n-capacity of E is called the conformal

capacity of E and denoted by capE.

A

C

Figure 7: Condenser E = (A,C).

Lemma 4.20. [22, 7.9] For all condensers (A,C) in Rn

(4.21) cap(A,C) = M(∆(C, ∂A; A)

).


Sets of zero capacity. A compact set E in Rn is said to be of capacity zero,

denoted capE = 0, if there exists a bounded set A with E ⊂ A and cap(A,E) =0. A compact set E ⊂ R

n, E 6= R

nis said to be of capacity zero if E can

be mapped by a Mobius transformation onto a bounded set of capacity zero.Otherwise E is said to be of positive capacity, and we write capE > 0.

Spherical symmetrizations. Let L be a ray from x0 to ∞ and E ⊂ Rn

bea compact set. We define spherical symmetrization of E in L as the set E

∗

satisfying the following conditions:

(1) x0 ∈ E∗ if and only if x0 ∈ E,

(2) ∞ ∈ E∗ if and only if ∞ ∈ E,

(3) For r ∈ (0,∞) the set E∗∩S

n−1(x0, r) is a closed spherical cap centered onL with the same (n− 1)-dimensional Lebesgue measure as E ∩ S

n−1(x0, r)for E ∩ S

n−1(x0, r) 6= ∅ and ∅ otherwise.

We note that E∗ is always compact and connected if E is.

L

E

LE

*

Figure 8: Spherical symmetrization.

Theorem 4.22. If E∗ is the spherical symmetrization of E in a ray L, then

(1) m(E∗) = m(E), and

(2) mn−1(∂E∗) ≤ mn−1(∂E).

Proof. (Outline, [7, p.224]) By Fubini’s theorem

m(E∗) =

∫ ∞

0

mn−1(E∗∩S

n−1(x0, r))dr =

∫ ∞

0

mn−1(E∩Sn−1(x0, r))dr = m(E),

which gives the first part.

To prove the second part, assume first that E is a polyhedron. Then forr ∈ (0,∞) the Brunn–Minkowski inequality yields

E∗(r) = x : d(x,E

∗) ≤ r ⊂ x : d(x,E) ≤ r∗ = E(r)∗,


and hence

mn−1(∂E∗) ≤ lim sup

r→0

m(E∗(r)) − m(E∗)

2r

≤ lim supr→0

m(E(r)) − m(E)

2r= mn−1(∂E).

The result for the general domains is obtained by approximating the boundarywith polyhedrons.

LL 0

CC

1

01

0 0L L 1

C1C0

0

**

Figure 9: Spherical symmetrization of a ring.

Theorem 4.23. If R = R(C0, C1) is a ring and if C∗0

and C∗1

are the sphrerical

symmetrizations of C0 and C1 in opposite rays L0, L1, then R∗ = R(C∗

0, C

∗1) is a

ring with cap R∗ ≤ cap R.

Proof. (Idea, [7, p.225]) Let u be a locally lipschitz function that is admissiblefor R. Choose u

∗ such that x : u∗(x) ≤ t = x : u(x) ≤ t∗. Then u

∗ isadmissible for R

∗ and from Theorem 4.22 we obtain

cap(R∗) ≤∫

Rn

|∇u∗|ndm ≤

∫

Rn

|∇u|ndm.

By taking the infimum over all admissible u the claim follows.

Canonical ring domains. The complementary components of the Grotzsch

ring RG,n(s) in Rn are B

nand [se1,∞], s > 1, and those of the Teichmuller ring

RT,n(s) are [−e1, 0] and [se1,∞], s > 0. We define two special functions γn(s),s > 1 and τn(s), s > 0 by

γn(s) = M

(∆(B

n, [se1,∞])

)= γ(s),

τn(s) = M(∆([−e1, 0], [se1,∞])

)= τ(s),

respectively. The subscript n is omitted if there is no danger of confusion. Weshall refer to these functions as the Grotzsch capacity and the Teichmuller ca-

pacity.


Γ

Bn

∞

se1

0

∞

Γ

se−e 11

Figure 10: Grotzsch ring RG,n(s) (left) and Teichmuller ring RT,n(s) (right).

Lemma 4.24. [22, 5.53] For all s > 1

γn(s) = 2n−1τn(s2 − 1)

and that τn : (0,∞) → (0,∞) is a decreasing homeomorphism.

Proof. (Idea) Apply Lemma 4.18 and an auxiliary Mobius transformation.

Lemma 4.25. [22, 5.63(1)] Let s > 0. Then

τ(s) ≤ γ(1 + 2s) = 2n−1τ(4s2 + 4s)

Proof. Let Γ = ∆(Sn−1(−e1/2, 1/2), [se1,∞]). Then by Lemma 4.24

M(Γ) = γ(1 + 2s) = 2n−1τ(4s2 + 4s).

By Lemma 4.4 τ(s) ≤ M(Γ).

Lemma 4.26. [3, (8.65),(8.62)] The following estimates hold for τn(t), t > 0:

τn(t) ≥ 21−nωn−1

(log

(λn

2(√

1 + t +√

t)))1−n

,

and for γn(1/r), r ∈ (0, 1):

γn(1/r) ≥ ωn−1

(log

λn

(1 +

√1 − r2

)

2r

)1−n

≥ ωn−1

(log

λn

r

)1−n

,

where λn is the Grotzsch ring constant depending only on n.

The value of λn is known only for n = 2, namely λ2 = 4. For n ≥ 3 it isknown that 20.76(n−1) ≤ λn ≤ 2en−1. For more information on λn, see [3, p.169].

Lemma 4.27. (see [9, 2.31]) Let 0 < r0 < 1. Then

M(∆(Bn(r), Sn−1)

)≥ γn(1/r) ≥ C(n, r0)M

(∆(Bn(r), Sn−1)

),

for r0 > r > 0.

Proof. By Lemma 4.26,

γn(1/r) ≥ ωn−1

(log

λn

(1 +

√1 − r2

)

2r

)1−n

≥ ωn−1

(log

λn

r

)1−n

.


We note that

logλn

r≤

(1 − log λn

log r0

)(log

1

r

),

for 0 < r < r0. Thus

γn(1/r) ≥ C(n, r0)ωn−1

(log

1

r

)1−n

= C(n, r0)M(∆(Bn(r), Sn−1)

),

with

C(n, r0) =

(1 − log λn

log r0

)1−n

.

The second inequality follows immediately from the fact that the line segment[0, r) is contained in the ball of radius r.

Remark 4.28. Note that C(n, r0) → 1 as r0 → 0 in Lemma 4.27.

Lemma 4.29. [22, 7.34] Let R = R(E,F ) be a ring in Rn, and let a, b ∈ E,

c,∞ ∈ F be distinct points. Then

M(∆(E,F )) ≥ τ

( |a − c||a − b|

).

Equality holds for E = [−e1, 0], a = 0, b = −e1, F = [se1,∞), c = se1, d = ∞.

It is not obvious from the definition how M(∆(E,F )), for nonempty E,F ∈R

n, depends on the geometric setup and the structure of the sets E,F . The

following lemma gives a lower bound for M(∆(E,F )) in the terms ofd(E,F )/ mind(E), d(F ).Lemma 4.30. [22, 7.38] Let E,F be disjoint continua in R

n with d(E), d(F ) > 0.Then

M(∆(E,F )) ≥ τ(4s2 + 4s) ≥ cn log(1 + 1/s)

where s = d(E,F )/ mind(E), d(F ) and cn > 0 is a constant depending only on

n.

This result can be improved to the following Lemma, which shows that M(∆(E,F ))and s = d(E,F )/ mind(E), d(F ) are simultaneously small or large, providedthat E,F are connected.

Lemma 4.31. [9, 2.30] For n ≥ 2 there are homeomorphisms h1, h2 of the

positive real axis with the following property. If E,F are the components of the

complements of a nondegenerate ring domain in Rn, then

h1(s) ≤ M(∆(E,F )) ≤ h2(s),

where s = d(E,F )/ mind(E), d(F ).


Spherical metric. The stereographic projection π : Rn → S

n(1

2en+1,

1

2) is de-

fined by

(4.32) π(x) = en+1 +x − en+1

|x − en+1|2, x ∈ R

n; π(∞) = en+1.

Stereographic projection is the restriction to Rn

of the inversion in Sn(en+1, 1)

in Rn+1

. Since π−1 = π, it follows that π maps the Riemann sphere S

n(1

2en+1,

1

2)

onto Rn. The chordal metric q in R

nis defined by

(4.33) q(x, y) = |π(x) − π(y)|; x, y ∈ Rn.

Lemma 4.34. [22, 7.37] If R = R(E,F ) is a ring, then

M(∆(E,F )) ≥ τ

(1

q(E)q(F )

),(4.35)

M(∆(E,F )) ≥ τ

(4q(E,F )

q(E)q(F )

).(4.36)

5. Quasiconformal mappings

A homeomorphism f : G → Rn, n ≥ 2, of a domain G in R

n is called qua-

siconformal if f is in ACLn, and there exists a constant K, 1 ≤ K < ∞ such

that

|f ′(x)|n ≤ K|Jf (x)|, |f ′(x)| = max|h|=1

|f ′(x)h|,

a.e. in G, where f′(x) is the formal derivative. The smallest K ≥ 1 for which

this inequality is true is called the outer dilatation of f and denoted by KO(f).If f is quasiconformal, then the smallest K ≥ 1 for which the inequality

|Jf (x)| ≤ Kl(f ′(x))n, l(f ′(x)) = min

|h|=1

|f ′(x)h|,

holds a.e. in G is called the inner dilatation of f and denoted by KI(f). Themaximal dilatation of f is the number K(f) = maxKI(f), KO(f). If K(f) ≤K, f is said to be K-quasiconformal. It is well-known that

KI(f) ≤ Kn−1

O (f), KO(f) ≤ Kn−1

I (f),

and hence KI(f) and KO(f) are simultaneously finite.

Theorem 5.1. [21, 32.2,33.2] Let f : G → Rn be a quasiconformal mapping.

Then

(1) f is differentiable a.e.,

(2) f satisfies condition (N), i.e. if A ⊂ G and m(A) = 0, then m(fA) = 0.

The next lemma gives another definition of quasiconformality. This definitionis called the geometric definition, and it is very useful in applications. The prooffor equivalence of these definitions is given in [21].


Lemma 5.2. A homeomorphism f : G → G′ is K-quasiconformal if and only if

M(Γ)/K ≤ M(f(Γ)) ≤ KM(Γ)

for every path family Γ in G.

We may also give geometric definitions for the inner and outer dilatations.Again, we refer to [21] for the proofs for the equivalence of these definitions.

Let G, G′ be domains in R

nand f : G → G

′ be a homeomorphism. Theninner and outer dilatations of f are respectively

KI(f) = supM(f(Γ))

M(Γ), KO(f) = sup

M(Γ)

M(f(Γ)),

where the suprema are taken over all path families Γ in G such that M(Γ) andM(f(Γ)) are not simultaneously 0 or ∞. The maximal dilatation of f is

K(f) = maxKI(f), KO(f).Theorem 5.3. [21, 13.2] Let f : G

′ → G′′, g : G → G

′ be quasiconformal map-

pings. Then

(1) KI(f−1) = KO(f),

(2) KO(f−1) = KI(f),(3) K(f−1) = K(f),(4) KI(f g) ≤ KI(f)KI(g),(5) KO(f g) ≤ KO(f)KO(g),(6) K(f g) ≤ K(f)K(g).

Examples. (see [21, pp.49-50]) (1) A homeomorphism f : G → fG satisfying

|x − y|/L ≤ |f(x) − f(y)| ≤ L|x − y|for all x, y ∈ G is called L-bilipschitz. It is easy to see that L-bilipschitz mapsare L

2(n−1) -quasiconformal.

(2) Let a 6= 0 be a real number, and let f(x) = |x|a−1x. We can extend f to

a homeomorphism f : Rn → R

nby defining f(0) = 0, f(∞) = ∞ for a > 0 and

f(0) = ∞, f(∞) = 0 for a < 0. Then f is quasiconformal with

KI(f) = |a|, KO(f) = |a|n−1 if |a| ≥ 1,KI(f) = |a|1−n

, KO(f) = |a|−1 if |a| ≤ 1.

(3) Let (r, ϕ, z) be the cylindrical coordinates of a point x ∈ Rn, i.e. r ≥ 0,

0 ≤ ϕ ≤ 2π, z ∈ Rn−2, and

x1 = r cos ϕ,

x2 = r sin ϕ,

xj = zj−2 for 3 ≤ j ≤ n.

The domain Gα, defined by 0 < ϕ < α, is called a wedge of angle α, α ∈ (0, 2π).Let 0 < α ≤ β < 2π. The folding f : Gα → Gβ, defined by

f(r, ϕ, z) = (r, βϕ/α, z),

is quasiconformal with KI(f) = β/α, KO(f) = (β/α)n−1.


6. An application of the modulus technique

As an application, we give a bound for how close to a point α the valuesattained by a quasiconformal mapping on a sequence of continua approachingthe boundary can be. The bound is given in the terms of the diameter of thecontinua involved. In order to prove this result, we need the following lemmas.This result is presented in [17, pp.638–639].

Lemma 6.1. Let w > 0 and t ∈ (0, minw2, 1/w). Then

1

2log

1

t< log

w

t< 2 log

1

t.

Proof. Since t < w2, we have 1/

√t < w/t. On the other hand, t < 1/w, or

w < 1/t, and hence w/t < 1/t2. By taking logarithm the claim follows.

Lemma 6.2. Let C ⊂ Bn be connected and 0 < d(C) ≤ 1. Then m ≡d(0, C)/d(C) < ∞ and if m > 0, then

M(Γ) ≥ 1

2τ(4m2 + 4m) ≥ 2−n

τ(m); Γ = ∆(Bn(1/2), C;Bn).

Proof. The second inequality holds by Lemma 4.25. To prove the first inequality,we note that if C ∩B

n(1/2) 6= ∅, then M(Γ) = ∞ and there is nothing to prove.

In what follows we may assume that C ∩ Bn(1/2) = ∅. Now the result follows

from the symmetry property of the modulus Lemma 4.11 and Lemma 4.30.

Theorem 6.3. Let f : Bn → Rn be a quasiconformal mapping or constant, α ∈

Rn and Cj a sequence of nondegenerate continua such that Cj → ∂Bn and |f(x)−

α| < Mj when x ∈ Cj, where Mj → 0 as j → ∞. If

lim supj→∞

τ

(1

d(Cj)

)(log

1

Mj

)n−1

= ∞,

then f ≡ α. In particular, if

lim supj→∞

(log

1

d(Cj)

)1−n(log

1

Mj

)n−1

= ∞,

then f ≡ α.

Proof. Suppose that f is not constant. Let Γj = ∆(Bn(1/2), Cj;Bn). Then by

Lemma 6.2

M(Γj) ≥ 2−nτ

(d(0, Cj)

d(Cj)

)≥ 2−n

τ

( 1

d(Cj)

).

Let w = d(fBn(1/2), α) > 0. Now by Lemma 4.14

M(fΓj) ≤ ωn−1

(log

w

Mj

)1−n

≤ ωn−1

(1

2log

1

Mj

)1−n

,


whenever Mj < minw2, 1/w by Lemma 6.1. Because M(Γj) ≤ K(M(fΓj)), the

estimates above yield

τ

(1

d(Cj)

)(log

1

Mj

)n−1

≤ 22n−1Kωn−1,

proving the first part of the claim.

The estimate (4.26) yields

τ(t) ≥ 21−nωn−1

[log

(λn

2

(√1 + t +

√t

))]1−n

where t = 1/d(Cj). It follows that[

log(

λn

2

(√1 + t +

√t

))]1−n

≥[

log(

λn

2(1 + 2

√t)

)]1−n

=

[log

(λn

2

(1 +

2√d(Cj)

))]1−n

.

We note that[

log(

λn

2

(1 +

2√d(Cj)

))]1−n

≥[2 log

( λn√d(Cj)

)]1−n

whenever j is large enough. Let v = λn. Now by Lemma 6.1[2 log

(λn√d(Cj)

)]1−n

≥(

2 log1

d(Cj)

)1−n

,

for√

d(Cj) < minv2, 1/v. Hence

τ

(1

d(Cj)

)(log

1

Mj

)n−1

≤ 22−2nωn−1

(log

1

d(Cj)

)1−n(log

1

Mj

)n−1

,

which gives the second part of the claim.

References

1. L. V. Ahlfors: Lectures on quasiconformal mappings, Van Nostrand Mathematical Stud-ies, No. 10 D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966.

2. C. Andreian–Cazacu: Foundations of quasiconformal mappings, Handbook of complexanalysis: geometric function theory, Vol. 2, edited by R. Kuhnau, 687–753, Elsevier, Am-sterdam, 2005.

3. G. D. Anderson, M. K. Vamanamurty and M. Vuorinen: Conformal invariants,

inequalities and quasiconformal mappings, Wiley-Interscience, 1997.4. P. Caraman: n-dimensional quasiconformal (QCf) mappings, Editura Academiei

Romane, Bucharest; Abacus Press, Newfoundland, N.J., 1974.5. F. W. Gehring: Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961)

499–519.6. F. W. Gehring: The Caratheodory convergence theorem for quasiconformal mappings in

the space, Ann. Acad. Sci. Fenn. Ser. A I 336/11, 1-21, 1963.7. F. W. Gehring: Quasiconformal mappings, Complex analysis and its applications (Lec-

tures, Internat. Sem., Trieste, 1975), Vol. II, 213–268, IAEA, Vienna, 1976.


8. F. W. Gehring: Quasiconformal mappings in Euclidean spaces, Handbook of complexanalysis: geometric function theory, Vol. 2, edited by R. Kuhnau, 1–29, Elsevier, Amster-dam, 2005.

9. V. Heikkala: Inequalities for conformal capacity, modulus and conformal invariants,Ann. Acad. Sci. Fenn. Math. Dissertationes 132 (2002), 1–62.

10. J. Heinonen: Lectures on analysis on metric spaces, Universitext, Springer-Verlag, NewYork, 2001.

11. J. Heinonen and P. Koskela: Definitions of quasiconformality, Invent. Math. 120(1995), no. 1, 61–79.

12. J. Heinonen and P. Koskela: Quasiconformal maps in metric spaces with controlled

geometry, Acta Math. 181 (1998), no. 1, 1–61.13. T. Iwaniec and G. Martin: Geometric function theory and non-linear analysis, Oxford

Mathematical Monographs, The Clarendon Press, Oxford University Press, New York,2001.

14. R. Kuhnau (ed.): Handbook of complex analysis: geometric function theory, Vol. 1–2,Amsterdam : North Holland/Elsevier, 2002, 2005.

15. O. Lehto: Univalent functions and Teichmller spaces, Graduate Texts in Mathematics,109. Springer-Verlag, New York, 1987.

16. O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane, Second edition.Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wis-senschaften, Band 126. Springer-Verlag, New York-Heidelberg, 1973.

17. A. Rasila: Multiplicity and boundary behavior of quasiregular maps, Math. Z. 250 (2005),611–640.

18. Yu. G. Reshetnyak: Space mappings with bounded distortion, Translations of Mathe-matical Monographs, 73, American Mathematical Society, Providence, RI, 1989.

19. S. Rickman: Quasiregular Mappings, Ergeb. Math. Grenzgeb. (3), Vol. 26, Springer-Verlag, Berlin, 1993.

20. J. Vaisala: On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I 298(1961), 1–36.

21. J. Vaisala: Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes inMath., Vol. 229, Springer-Verlag, Berlin, 1971.

22. M. Vuorinen: Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math.,Vol. 1319, Springer-Verlag, Berlin, 1988.

23. M. Vuorinen (ed.): Quasiconformal space mappings. A collection of surveys 1960–1990,

Lecture Notes in Mathematics, 1508, Springer-Verlag, Berlin, 1992.

Antti Rasila E-mail: [email protected]:Helsinki University of Technology, Institute of Mathematics,

P.O.Box 1100, FIN-02015 HUT, Finland


The universal Teichmuller space and related topics

Toshiyuki Sugawa

Abstract. In this survey, we give an expository account of the universal Teichmuller

space with emphasis on the connection with univalent functions. In the theory, the

Schwarzian derivative plays an important role. We also introduce many interesting results

involving Schwarzian derivatives and pre-Schwarzian derivatives, as well.

Keywords. universal Teichmuller space, univalent function, Schwarzian derivative, pre-

Schwarzian derivative.

2000 MSC. Primary: 30F60, Secondary: 30C55, 30C62.

Contents

1. Preliminary 262

1.1. Quasiconformal mappings 262

1.2. Hyperbolic Riemann surfaces 264

1.3. Quadratic differentials 264

1.4. Univalent functions 266

1.5. Grunsky inequality 266

1.6. Schwarzian derivative 268

2. The universal Teichmuller space 269

2.1. Definition 1: the quotient space of quasiconformal maps 269

2.2. Definition 2: quasisymmetric functions 269

2.3. Definition 3: marked quasidisks 270

2.4. Definition 4: Bers embedding 271

2.5. Equivalence of T1 through T4 271

3. Analytic properties of the Bers embedding 272

3.1. The Teichmuller space of a Riemann surface 272

3.2. Relationship with quasi-Teichmuller spaces 274

3.3. The Bers projection 275

3.4. Convexity 275

3.5. Teichmuller distance and other natural distances (metrics) 276

262 T. Sugawa IWQCMA05

4. Pre-Schwarzian models 277

4.1. The models T (D) and T (H) 277

4.2. The model T (D∗) 278

4.3. Loci of typical subclasses of S 279

5. Univalence criteria 280

5.1. Univalence criteria due to Nehari and Ahlfors-Weill 280

5.2. Inner radius and outer radius 281

5.3. Pre-Schwarzian counterpart 282

5.4. Directions of further investigation 283

References 284

1. Preliminary

In this section, we prepare basic tools to understand the universal Teichmuller space.

The material is more or less standard, but for convenience, an expository account will

be given without proofs. The most convenient reference for overall topics is perhaps the

recently published handbook [61].

1.1. Quasiconformal mappings. A homeomorhism f of a plane domain D onto an-

other domain D′ is called a quasiconformal map if f has locally square integrable partial

derivatives (in the sense of distribution) and satisfies the inequality

|fz| ≤ k|fz|almost everywhere in D, where k is a constant with 0 ≤ k < 1,

fz = 1

2(fx − ify), fz = 1

2(fx + ify)

and

fx =∂f

∂x, fy =

∂f

∂y.

It turns out that f preserves sets of (2-dimensional) Lebesgue measure zero and, in par-

ticular, fz 6= 0 a.e. Thus the quotient µ = fz/fz is well defined as a Borel measurable

function on D and satisfies ‖µ‖∞ ≤ k < 1. This function is sometimes called the complex

dilatation of f and denoted by µf . More specifically, f is also called a K-quasiconformal

map, where K = (1 + k)/(1− k). The minimal K = (1 + ‖µ‖∞)/(1− ‖µ‖∞) is called the

maximal dilatation of f and denoted by K(f). It is known that a 1-quasiconformal map is

conformal (i.e., biholomorphic) and vice versa. The composition of a K1-quasiconformal

map and a K2-quasiconformal map is K1K2-quasiconformal map and the inverse map of

a K-quasiconformal map is also K-quasiconformal. In particular, K-quasiconformality is

The universal Teichmuller space and related topics 263

preserved under composition with conformal maps. Therefore, K-quasiconformality and,

hence, quasiconformality can be defined for homeomorphisms between Riemann surfaces.

In particular, we can argue quasiconformality of a homeomorphism of the Riemann sphere

C = C ∪ ∞.More precise information about compositions of quasiconformal maps will be needed

later. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then the complex

dilatation of g f−1 is given by

(1.1.1) (µgf−1 f)fz

fz

=µg − µf

1 − µf · µg

.

In particular, we obtain the following lemma.

Lemma 1.1.2. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then g f−1 is

conformal on Ω′ if and only if µf = µg a.e. in Ω.

Fundamental in the theory of quasiconformal maps is the following existence and

uniqueness theorem.

Theorem 1.1.3 (The measurable Riemann mapping theorem). For any measurable

function µ on C with ‖µ‖∞ < 1, there exists a unique quasiconformal map f : C → C

such that f(0) = 0, f(1) = 1 and fz = µfz a.e. in C.

For the proof of the theorem and for more information about quasiconformal maps,

the reader should consult the books [3] and [69] as well as the paper [4] by Ahlfors and

Bers. See also the article “Beltrami Equation”, by Srebro and Yakubov, in [61, vol. 2] for

the recent development.

We denote by Belt(D) the open unit ball of the space L∞(D) for a domain (or, more

generally, a measurable set) D. An element µ of Belt(D) is called a Beltrami coefficient

on D. For a Beltrami coefficient µ on C, the function f given in the measurable Riemann

mapping theorem will be denoted by fµ throughout the present survey.

Let µ be a Beltrami coefficient on the outside D∗ of the unit disk. We extend µ to

µ∗ ∈ Belt(C) by setting µ

∗(z) = µ(1/z) for z ∈ D. Let f be a quasiconformal auto-

morphism of C fixing 1,−1,−i with µf = µ∗. Since f(z) and 1/f(1/z) both have the

same complex dilatation µ∗ and satisfy the same normalization condition, they must be

equal by uniqueness part of the measurable Riemann mapping theorem. In particular,

|f(z)|2 = 1 for z ∈ ∂D, and consequently, f maps D∗ onto itself. We define wµ : C → C

by wµ = f. Recall that wµ fixes 1,−1 and −i.The following fact was observed by Ahlfors and Bers [4].


Theorem 1.1.4. Let µt be a family of Beltrami coefficients on C holomorphically param-

eterized over a complex manifold X. Then the map t 7→ fµt(z) is holomorphic on X for a

fixed z ∈ C.

We do not explain the meaning of “holomorphically parameterized” here. It is, however,

sufficient practically to know that (tµ + ν)/(1 + tνµ) is a family of Beltrami coefficients

holomorphically parameterized over the unit disk |t| < 1, where µ, ν ∈ Belt(C).

1.2. Hyperbolic Riemann surfaces. A connected complex manifold of complex di-

mension one is called a Riemann surface. The Poincare-Koebe uniformization theorem

tells us that every Riemann surface R admits an analytic universal covering projection

p of the unit disk D = z ∈ C : |z| < 1 onto R except for the case when R is confor-

mally equivalent to the Riemann sphere C, the complex plane C, the punctured complex

plane C∗ = C \ 0 or a complex torus (a smooth elliptic curve). Those non-exceptional

Riemann surfaces are called hyperbolic.

The group of analytic automorphisms of R is denoted by Aut(R). The group of disk

automorphisms Aut(D) is identified with PSU(1, 1) and isomorphic to PSL(2,R) through

the Mobius transformation M : H = z : Im z > 0 → D defined by M(z) = (z−i)/(z+i).Thus Aut(D) inherits a structure of real Lie group. A subgroup Γ of Aut(D) is called

Fuchsian if Γ is discrete. It is known that Γ is discrete if and only if Γ acts on D properly

discontinuously. Note also that Γ is torsion-free if and only if Γ acts on D without fixed

points. The covering transformation group Γ = γ ∈ Aut(D) : p γ = p is a torsion-

free Fuchsian group and will be called the Fuchsian model of R. Conversely, for a given

torsion-free Fuchsian group Γ the quotient space D/Γ has natural complex structure so

that the projection D → D/Γ becomes an analytic universal covering. In this way, the

theory of hyperbolic Riemann surfaces can be translated into that of torsion-free Fuchsiangroups.

Since the Poincare metric ρD(z)|dz| = |dz|/(1−|z|2) is invariant under the pull-back by

analytic automorphisms of D, it projects to a smooth metric, denoted by ρR = ρR(w)|dw|,on the hyperbolic Riemann surface R via p. The metric ρR is called the hyperbolic metric

of R. Thus ρR is characterized by the relation ρD = p∗(ρR) = ρR(p(z))|p′(z)||dz|.

Note that ρR has constant Gaussian curvature −4, in other words, ∆ log ρR = 4ρ2

R.

The Schwarz-Pick lemma implies the contraction property f ∗ρS ≤ ρR for a holomorphic

map f : R → S between hyperbolic Riemann surfaces R and S, where equality holds at

some (hence every) point in R iff f is a covering projection of R onto S.

1.3. Quadratic differentials. Let H(D) be the set of analytic functions on the unit

disk D and let n be a non-negative integer. For a Fuchsian group Γ, a function ϕ ∈ H(D)

is said to be automorphic for Γ (with weight −2n) if ϕ satisfies the functional equation


(ϕ γ)(γ′)n = ϕ for every γ ∈ Γ, that is to say, ϕ(z)dzn is an invariant n-form for Γ. The

set of automorphic functions for Γ with weight −2n will be denoted by Hn(D,Γ).

An element ϕ of Hn(D,Γ) for a torsion-free Fuchsian group Γ projects to a holomorphic

n-form q = q(w)dwn on R = D/Γ so that p∗nq = ϕ(z)dzn, where p∗nq means the pull-back

q(p(w))(p′(w))n of the n-form q by the canonical projection p : D → D/Γ.

We now define two norms for ϕ ∈ Hn(D,Γ) by

‖ϕ‖An(D,Γ) =

∫∫

ω

|ϕ(z)|(1 − |z|2)n−2dxdy,

‖ϕ‖Bn(D,Γ) = supz∈D

|ϕ(z)|(1 − |z|2)n,

where ω is a fundamental domain for Γ, that is, a subdomain of D such that ω∩γ(ω) = ∅for every γ ∈ Γ with γ 6= id,

⋃γ∈Γ

γ(ω) = D and ∂ω is of zero area. We denote by

An(D,Γ) and Bn(D,Γ) the subsets of Hn(D,Γ) consisting of ϕ with finite norm ‖ϕ‖An(D,Γ)

and ‖ϕ‖Bn(D,Γ), respectively. It is easy to see that these become complex Banach spaces.

When Γ is the trivial group 1, we write An(D) and Bn(D) for An(D, 1) and Bn(D, 1),

respectively.

The definition of the spaces An(D) and Bn(D) can be extended for hyperbolic Riemann

surfaces R. Let Hn(R) denote the set of holomorphic n-forms on R and set

‖ϕ‖An(R) =

∫∫

R

|ϕ(w)|ρR(w)2−ndxdy,

‖ϕ‖Bn(R) = supw∈R

|ϕ(w)|ρR(w)−n

for ϕ = ϕ(w)dwn in Hn(R). Here, we should note that |ϕ(w)|ρR(w)−n does not depend on

the choice of the local coordinate w, in other words, |ϕ|ρ−nR can be regarded as a function

on R.

The Banach spaces An(D,Γ) and An(D/Γ) (resp. Bn(D,Γ) and Bn(D/Γ)) are isomet-

rically isomorphic through the pull-back p∗n by the projection p : D → D/Γ. Also, the

following invariance property is convenient to note.

Lemma 1.3.1. Let R and S be hyperbolic Riemann surfaces and let p : R → S be a

conformal homeomorphism. Then the pullback operator p∗n : Bn(S) → Bn(R) is a linear

isometry, in other words,

‖p∗nϕ‖Bn(R) = ‖ϕ‖Bn(S), ϕ ∈ Bn(S).

In the theory of Teichmuller spaces, it is important to consider the spaces A2 and B2

as we shall see later. A 2-form q(w)dw2 is traditionally called a quadratic differential.


1.4. Univalent functions. In connection with the universal Teichmuller space, the the-

ory of univalent functions is of particular importance. The best textbook in this direction

is [67] by O. Lehto.

We denote by S the set of analytic univalent functions f on the unit disk so normalized

that f(0) = 0 and f′(0) = 1. An analytic function f around the origin is said to be

strongly normalized if f(0) = f′(0)−1 = f

′′(0) = 0. Let S0 be the subset of S consisting

of strongly normalized functions. For f ∈ S , the function g = f/(1 + af), where

a = f′′(0)/2, is strongly normalized but not necessarily analytic in D. It is thus natural

to consider the wider class

S0 = f : meromorphic and univalent in D and strongly normalized

than S0.

The following meromorphic counterpart is also useful in the theory of univalent func-

tions. Let Σ be the set of meromorphic univalent functions F on the exterior D∗ = ζ ∈

C : |ζ| > 1 of the unit disk so normalized that

(1.4.1) F (ζ) = ζ + b0 +b1

ζ+b2

ζ2+ . . .

in |ζ| > 1.

For f ∈ S , the function F (ζ) = 1/f(1/ζ) belongs to Σ and satisfies the condition

0 /∈ F (D∗), and vice versa. Let Σ′ denote the set of those functions F ∈ Σ that satisfy

0 /∈ F (D∗). Moreover, b0 = 0 for a function F (ζ) = ζ + b0 + b1/ζ + . . . in Σ if and only if

f ∈ S0, where f(z) = 1/F (1/z). Hence, if we set

Σ0 = F ∈ Σ : F (ζ) − ζ → 0 as ζ → ∞,

the correspondence f(z) 7→ F (ζ) = 1/f(1/ζ) gives bijections of S0 onto Σ0 and of S0

onto Σ′0, where we define Σ′

0= Σ0 ∩ Σ′

.

1.5. Grunsky inequality. For a meromorphic function F near the point at infinity with

an expansion of the form (1.4.1), we take a single-valued branch of log((F (ζ)−F (ω))/(ζ−ω)) in |ζ| > R and |ω| > R for sufficiently large R > 0 and expand it in the form

logF (ζ) − F (ω)

ζ − ω= −

∞∑

j=1

∞∑

k=1

bj,k

ζjωk

there. The coefficients bj,k are called the Grunsky coefficients of F. It is easy to see that

bj,k = bk,j and b1,k = bk for j, k ≥ 1, where bk is the coefficient in (1.4.1). The last relation

is deduced in the following way. If we write F (ζ) = ζ + b0 +G(ζ), then G(ζ) = O(|ζ|−1)


as ζ → ∞. Fix ω for a moment. Since

logF (ζ) − F (ω)

ζ − ω= log

(1 +

G(ζ) −G(ω)

ζ − ω

)=G(ζ) −G(ω)

ζ − ω+O(|ζ|−2),

we obtain∞∑

k=1

b1,k

ωk= − lim

ζ→∞ζ log

F (ζ) − F (ω)

ζ − ω

= − limζ→∞

ζG(ζ) −G(ω)

ζ − ω

= G(ω),

from which the required relation follows.

The following theorem is greatly useful in the theory of Teichmuller spaces as well as

the theory of univalent functions. See [42], [29] or [86] for the proof and applications.

Theorem 1.5.1 (Grunsky). A meromorphic function F (ζ) with expansion of the form

(1.4.1) around ζ = ∞ is analytically continued to a univalent meromorphic function in

|ζ| > 1 if and only if the inequality

(1.5.2)∞∑

k=1

k

∣∣∣∣∣

∞∑

j=1

bj,kxj

∣∣∣∣∣

2

≤∞∑

j=1

|xj|2j

holds for an arbitrary sequence of complex numbers x1, x2, . . . .

The inequality in (1.5.2) is known as the strong Grunsky inequality. Noting b1,k = bk,

we take (x1, x2, x3, . . . ) = (1, 0, 0, . . . ) to obtain

(1.5.3)∞∑

k=1

k|bk|2 ≤ 1.

This inequality is known as Gronwall’s area theorem.

It is also known that inequality (1.5.2) can be replaced in the above theorem by the

(classical) Grunsky inequality:

(1.5.4)

∣∣∣∣∣

∞∑

j=1

∞∑

k=1

bj,kxjxk

∣∣∣∣∣ ≤∞∑

j=1

|xj|2j.

The symmetric matrix (√jkbj,k) defines a linear operator on ℓ

2, where bj,k are the

Grunsky coefficient of a meromorphic function F (ζ) around ζ = ∞. This is sometimes

called the Grunsky operator and will be denoted by G[F ] in the following. The strong


Grunsky inequality says that F ∈ Σ if and only if G[F ] is a bounded linear operator on

ℓ2 with operator norm ≤ 1. Here, the operator norm ‖G[F ]‖ of G[F ] is defined by

‖G[F ]‖2 = sup‖y‖2=1

∞∑

k=1

∣∣∣∣∣

∞∑

j=1

√jkbj,kyj

∣∣∣∣∣

2

,

where ‖y‖2 = (∑

k |yk|2)1/2 for y = (y1, y2, . . . ). Thus, F ∈ Σ ⇔ ‖G[F ]‖ ≤ 1. It is

known (cf. [86]) that F has a quasiconformal extension to C if and only if ‖G[F ]‖ < 1.

See also the article “Univalent holomorphic functions with quasiconformal extensions”, by

Krushkal, in [61, vol. 2].

1.6. Schwarzian derivative. For a non-constant meromorphic function f on a domain,

we define Tf and Sf by

Tf =f′′

f ′ = (log f ′)′,

Sf = (Tf )′ − 1

2(Tf )

2 =f′′′

f ′ − 3

2

(f′

f

)2

.

These are called the pre-Schwarzian derivative and the Schwarzian derivative of f, respec-

tively. Note that Tf is analytic at a finite point z0 if and only if f is analytic and injective

around z0. Similarly, Sf is analytic at z0 if and only if f is meromorphic and injective

around z0. The following two lemmas show usefulness of these operations.

Lemma 1.6.1. Let f be a non-constant meromorphic function on a domain D. The

pre-Schwarzian derivative of f vanishes on D if and only if f is (the restriction of)

a similarity. The Schwarzian derivative of f vanishes on D if and only if f is (the

restriction of) a Mobius transformation.

Lemma 1.6.2. Let f and g be non-constant meromorphic functions for which the com-

position f g is defined. Then

Tfg = (Tf ) g · g′ + Tg = g∗1(Tf ) + Tg,

Sfg = (Sf ) g · (g′)2 + Sg = g∗2(Sf ) + Sg.

Combining these lemmas, we observe that SLfM = M∗2(Sf ) for Mobius transforma-

tions L and M. Thus the Schwarzian derivative behaves like a quadratic differential.


2. The universal Teichmuller space

We have two choices to develop the theory of the (universal) Teichmuller space; the

unit disk model or the upper half-plane model. Although they can be translated into

each other, in principle, via the Mobius transformation z 7→ (z − i)/(z + i), both models

have their own advantage and thus can be chosen at will according to the purpose. In the

present survey, we will take the unit disk model to connect with the theory of univalent

functions in a direct way.

2.1. Definition 1: the quotient space of quasiconformal maps. We denote by

QC(D) the set of quasiconformal automorphisms of the unit disk D. As we will observe

later, every function in QC(D) extends to a unique homeomorphism of the closed unit

disk D. Thus, we may think that every f ∈ QC(D) is a self-homeomorphism of the closed

unit disk D. Two functions f and g in QC(D) are said to be Teichmuller equivalent and

denoted by fT∼g if there exists a disk automorphism L ∈ Aut(D) such that g = L f on

∂D. The quotient space QC(D)/T∼ is a model of the universal Teichmuller space and will

be denoted by T1 for a moment. The equivalence class represented by f ∈ QC(D) will be

denoted by [f ] below.

Let f, g ∈ QC(D). The Teichmuller distance between p = [f ] and q = [g] is defined by

d1(p, q) = inff1

T∼f,g1

T∼g

1

2logK(g1 f−1

1).

Recall here that K(f) denotes the maximal dilatation of f. By a compactness property

of quasiconformal maps, one can check that d1(p, q) is indeed a distance on T1. In this

way, T1 becomes a metric space. It can also be shown that T1 is a complete metric

space with metric d1 by a normality property of the set of normalized K-quasiconformal

automorphisms of C (see [69]).

2.2. Definition 2: quasisymmetric functions. The notion of quasisymmetric func-

tions was created by Beurling and Ahlfors [18] for functions on the real line. We give

here a corresponding definition of quasisymmetric functions on the unit circle. A sense-

preserving homeomorphism h of the unit circle ∂D is called quasisymmetric if

1

M≤ |h(ei(s+t)) − h(eis)|

|h(eis) − h(ei(s−t))| ≤M, s ∈ R, 0 < t <π

2

for a constant M ≥ 1. The set of all quasisymmetric functions on the unit circle will be

denoted by QS(∂D). The main result in [18] can be stated as follows.


Theorem 2.2.1 (Beurling-Ahlfors). The restriction of a quasiconformal automorphism

of the unit disk to the unit circle is quasisymmetric. Conversely, a quasisymmetric func-

tion on the unit circle can be extended to a quasiconformal automorphism of the unit disk.

Two functions h1 and h2 on the unit circle are called Mobius equivalent if there exists

a disk automorphism L ∈ Aut(D) such that h2 = Lh1. Let T2 denote the quotient space

of QS(∂D) by the Mobius equivalence. By the above theorem of Beurling and Ahlfors,

one readily sees that T1 can be identified with T2 in a natural manner.

In order to get rid of taking quotient, we can define T2 as follows. A (sense-preserving)

homeomorphism h of ∂D is said to be normalized if h fixes the points 1,−1 and −i. Since

every Mobius equivalence class of quasisymmetric functions is represented by a unique

normalized one, one can identify T2 with the set of normalized quasisymmetric functions

on the unit circle.

See [37] for a modern treatment of quasisymmetric functions. The survey article “Uni-

versal Teichmuller space”, by Gardiner and Harvey, in [61, vol. 1] puts emphasis on the

connection with quasisymmetric functions.

2.3. Definition 3: marked quasidisks. A simply connected domain D in C is called a

quasidisk if D is the image of the unit disk under a quasiconformal automorphism of C. If

D is the image under a K-quasiconformal automorphism, then D is called a K-quasidisk.

Many characteristic properties of quasidisks are known. See, for instance, [40].

Let D be a quasidisk (or a Jordan domain more generally) and x1, x2, x3 are positively

ordered (distinct) points on ∂D. The quadruple (D, x1, x2, x3) will be called a marked

quasidisk. By the Riemann mapping theorem and the Caratheodory extension theorem,

there exists a unique conformal homeomorphism g : H → D with g(0) = x1, g(1) = x2

and g(∞) = x3.

We denote by Q the set of all marked quasidisks in C. Two marked quasidisks (D, xj)

and (D′, x

′j) are said to be Mobius equivalent if D′ = L(D) and x

′j = L(xj), j = 1, 2, 3,

for some Mobius transformation L ∈ Mob = Aut(C). We can define a pseudo-metric on

Q by

d((D, xj), (D′, x

′j)) = ‖Sf‖B2(D),

where f is a conformal homeomorphisms of D onto D′ with f(xj) = x′j. It is easy to see

that d(D,D′) = 0 if and only if D and D′ are Mobius equivalent.

The set T3 of Mobius equivalence classes of all marked quasidisks constitutes another

model of the universal Teichmuller space and the above-defined pseudo-metric gives a

metric on T3, which will be denoted by d3, i.e.,

d3(p, q) = inf(D,xj)∈p,(D′,x′

j)∈qd((D, xj), (D

′, x

′j))


for p, q ∈ T3.

We can again take a suitable normalization to avoid the process of quotient and even

marking. For instance, we may say that a quasidisk D is normalized if its boundary

contains the points 0, 1 and ∞ in positive order along the boundary curve. If we denote

by Q0 the set of normalized quasidisks in C, then T3 can be identified with Q0 naturally,

and the restriction of the distance d on Q0 corresponds to the distance d3 on T3.

In the above, the marking is important. For two simply connected hyperbolic domains

D1 and D2, we set

d(D,D′) = inff :D→D′ conformal

‖Sf‖B2(D).

It is easy to see that d is a pseudo-distance. Lehto [67] posed a question whether or

not d(D,D′) = 0 implies that D and D′ are Mobius equivalent. Osgood and Stowe [82]

answered to this question in the negative (see also [19]).

2.4. Definition 4: Bers embedding. Let D be a hyperbolic domain in C. We define

a subset T (D) of B2(D) to consists of those holomorphic quadratic differentials ϕ(z)dz2

on D such that ϕ = Sf for some univalent meromorphic function f on D which extends

to a quasiconformal automorphism of the Riemann sphere. Note that ‖Sf‖B2(D) ≤ 12 for

every univalent meromorphic function f on D (see §5.2 and [10]). By Lemmas 1.6.1 and

1.6.2, for a Mobius transformation L, the pull-back L∗2

gives an isometric isomorphism of

B2(L(D)) onto B2(D). In particular, for a circle domain ∆, that is, the interior or the

exterior of a circle, or a half-plane, the space B2(∆) is isomorphic, say, to B2(D∗). The

space T4 = T (D∗) (or its equivalent) is a model of the universal Teichmuller space and

known as the Bers embedding of the universal Teichmuller space.

Ahlfors [2] showed the following.

Theorem 2.4.1. T (D∗) is a bounded, connected and open subset of B2(D∗).

Thus T4 = T (D∗) inherits a complex structure and a metric from B2(D∗). We denote by

d4 the distance, namely, d4(ϕ, ψ) = ‖ϕ− ψ‖B2(D∗) for ϕ, ψ ∈ T4. Since T (D∗) is bounded,

the distance d4 is not complete.

2.5. Equivalence of T1 through T4. We see now that the above definitions of the

universal Teichmuller space are all equivalent. Firstly, consider the restriction map

QC(D) → QS(∂D) defined by f 7→ f |∂D. Then this map yields a bijection of T1 onto

T2.

Secondly, we see the equivalence of T3 and T4. For ϕ ∈ T4 = T (D∗), by definition,

there exists a quasiconformal map f of C fixing 0, 1,∞ such that f is conformal on D∗

and satisfies Sf = ϕ. Then the image D = f(D∗) is a normalized quasidisk. Therefore, the

correspondence ϕ 7→ D gives a map T4 → T3. We next show that this map is bijective.


Suppose that a normalized quasidisk D is given. By definition, D = h(D∗) for some

quasiconformal map h of C with h(1) = 0, h(−1) = 1 and h(−i) = ∞. Let µ = µh|D∗

and set f = h (wµ)−1 : C → C. Then f is quasiconformal map of C, is conformal on

wµ(D∗) = D∗ and satisfies f(1) = 0, f(−1) = 1 and f(−i) = ∞. Therefore, ϕ = Sf

belongs to T4 = T (D∗). In this way, we obtain the map of T3 into T4, which is obviously

the inverse map of the previously defined map of T4 to T3. We have now concluded that

T3 and T4 are equivalent by those maps.

Finally, we connect T1 with T4. Let h ∈ QC(D). We define µ ∈ Belt(C) by

µ =

µh on D,

0 on D∗

and define a quasiconformal map f : C → C by f = fµ, where fµ was defined in §1.1.

Since f is conformal in D∗, the Schwarzian derivative Sf belongs to T4 = T (D∗). Note that

f h−1 is conformal in D by construction. Let h1 ∈ QC(D) be Teichmuller equivalent to

h and define f1 in the same way as above. We claim now that Sf1= Sf . By assumption,

h1 = L h on ∂D for an L ∈ Aut(D). Define a map g : C → C by

g =

f1 f−1 on C \ f(D),

f1 h−1

1 L h f−1 on f(D).

It is clear that g is conformal on f(D) and f(D∗). Furthermore, since h−1

1 L h = id

on ∂D, the map g is continuous on C. Since C = f(∂D) and g(C) = f1(∂D) are both

quasicircles, it turns out that g is quasiconformal in C. Since µg = 0 a.e., we conclude

that g is conformal, hence, a Mobius map. Because of the relation f1 = g f on f(D∗),

Sf = Sf1follows as required.

In this way, we obtain the mapping of T1 to T4 : [h] 7→ Sf |∗D. It is not difficult to see

that this mapping is bijective. This map is called the Bers embedding.

3. Analytic properties of the Bers embedding

3.1. The Teichmuller space of a Riemann surface. It is beyond the scope of the

present survey to develop the theory of Teichmuller spaces of Riemann surfaces in full

generality. Here, our focus will be on the Bers embedding of the Teichmuller space of a

Riemann surface. See [75], [45], [35], [36] for general properties of Teichmuller spaces. See

also [1], [113] for a differential geometric approach, [94] for an algebraic approach.

For simplicity, we assume a Riemann surface R to be hyperbolic, in other words,

there exists a torsion-free Fuchsian group Γ acting on D such that D/Γ is conformally

equivalent to R. Thus, we can identify R with D/Γ. We denote by p : D → D/Γ = R


the canonical projection. Two quasiconformal maps fj : R → Sj, j = 1, 2, are called

Teichmuller equivalent if there exists a conformal homeomorphism g : S1 → S2 such that

f−1

2 g f1 : R → R is homotopic to the identity relative to the ideal boundary. We omit

the explanation of the term “relative to the ideal boundary”. See the references given

above for details. Also, [33] gives several useful equivalent conditions for that.

The Teichmuller space Teich(R) of R is defined as the set of all the Teichmuller equiv-

alence classes of such quasiconformal maps of R onto another surface.

Suppose that f1 : R → S1 and f2 : R → S2 are quasiconformal maps. Let Γj be a

Fuchsian model of Sj acting on D and hj : D → D be a lift of fj, namely, pj hj = fj p,where pj : D → D/Γj = Sj is the canonical projection. Then, it is known that f1 and f2

are Teichmuller equivalent if and only if h1 and h2 are Teichmuller equivalent in the sense

of §2.1. Note that hj γ h−1

j ∈ Γj for each γ ∈ Γ, namely, hjΓh−1

j = Γj.

Set

QC(D,Γ) = h ∈ QC(D) : hΓh−1 is Fuchsian

and denote by Teich(Γ) the quotient space QC(D,Γ)/T∼. As we have seen, Teich(R) and

Teich(Γ) are canonically isomorphic through the universal covering projection p : D →D/Γ = R. Also, Teich(Γ) is naturally contained in Teich(1) = T1. In this sense, the

universal Teichmuller space T (D∗) contains all the Teichmuller space of an arbitrary

hyperbolic Riemann surface.

By using (1.1.1), the complex dilatation of f ∈ QC(D,Γ) is seen to be contained in

Belt(D,Γ) = µ ∈ Belt(D) : (µ γ)γ′/γ′ = µ ∀γ ∈ Γ.

Furthermore, for h ∈ QC(D,Γ), let f be the function constructed in §2.5 and let γ ∈ Γ.

Since f and γ f γ−1 has the same complex dilatation, γ f γ−1 = L f for an

L ∈ Aut(C) = Mob by Lemma 1.1.2. Lemma 1.6.2 now implies that γ∗2(Sf ) = Sf .

Therefore, Sf is contained in the closed subspace B2(D∗,Γ) of B2(D

∗) defined in §1.3. As

in the previous section, we see that Sf depends only on the Teichmuller equivalence class

of h in QC(D,Γ) and the corresponding h 7→ Sf is one-to-one, we obtain an embedding

βΓ : Teich(D,Γ) → B2(D∗,Γ), which is called the Bers embedding of Teich(D,Γ). We set

T (D∗,Γ) = βΓ(Teich(D,Γ)).

Bers [15] showed that T (D∗,Γ) is a bounded domain in B2(D

∗,Γ). It is obvious that

T (D∗,Γ) is contained in T (D∗) by definition. Indeed, by using the Douady-Earle extension

[28], it can be seen that

T (D∗,Γ) = T (D∗) ∩B2(D

∗,Γ)

and that T (D∗,Γ) is contractible.


3.2. Relationship with quasi-Teichmuller spaces. In view of the description of the

set T (D∗,Γ), it may be natural to consider the following sets more generally. Let D be a

hyperbolic domain in C and let G be a subgroup of Aut(D). Typically, G is a Kleinian

group and D is a connected component of its region of discontinuity. Then we set (cf. [98])

S(D,G) = ϕ ∈ B2(D,G) : ∃f : D → C s.t. ϕ = Sf and f is univalent in D,

T (D,G) = ϕ ∈ B2(D,G) : ∃f : D → C s.t. ϕ = Sf and f extends to a qc map of C,

For a circle domain ∆ and a Fuchsian group Γ acting on ∆, the set S(∆,Γ) sometimes

called the quasi-Teichmuller space of Γ. (But, note that this terminology is not popular.)

Clearly, T (D,G) ⊂ S(D,G). It is easy to see that S(D∗) is closed while, as Ahlfors

showed, T (D∗) is open in B2(D∗). The boundary of T (∆,Γ) in B2(∆,Γ) is called the Bers

boundary and is important in relation with the deformation theory of Kleinian groups

(see [16]).

When G is the trivial group 1, we write S(D), T (D) for S(D, 1), T (D, 1), respectively.

Note that under the mapping f 7→ Sf , the sets S0 and Σ0 correspond to S(D) and

S(D∗), respectively, in one-to-one fashion. It is a challenging problem to characterize

those functions f in S whose Schwarzian derivatives lie on ∂T (D). See [7] and [43] forsome attempts.

In 1970’s, it had been a conjecture of Bers [16] that the closure of T (D∗) in B2(D∗)

is S(D∗). In 1978, Gehring [39] disproved it. Prior to it, Gehring [38] proved the weaker

assertion that the interior of S(D∗) in B2(D∗) coincides with T (D∗). See [34] for a relevant

result. Thurston [110] proved the more striking result that S(D∗) even has an isolated

point in B2(D∗) (see also [5]). After that, the Bers conjecture was reformulated in the form

that the closure of T (D∗,Γ) is equal to S(D∗

,Γ) for a cofinite Fuchsian group Γ, that is, a

finitely generated Fuchsian group of the first kind. (This is nowadays generalized to the

Bers-Thurston density conjecture.) Shiga [95] proved a weaker version of it: the interior of

S(D∗,Γ) in B2(D

∗,Γ) coincides with T (D∗

,Γ) for a cofinite Γ. In the line of these studies,

the author showed that S(D∗,Γ) \ T (D∗) 6= ∅ for a Fuchsian group Γ of the second kind

([99]) and that the interior of S(D∗,Γ) in B2(D

∗,Γ) coincides with T (D∗

,Γ) for a finitely

generated, purely hyperbolic Fuchsian group Γ of the second kind ([100]). Matsuzaki [71]

generalized the former to the case of a certain kind of infinitely generated Fuchsian groups

of the first kind. In recent years, a huge amount of progress has been made in the theory

of Kleinian groups, which enabled to prove the Bers-Thurston conjecture partially. See,

for instance, [20] and [80] for partial solutions to the conjecture.

We end the subsection with the following conjecture.


Conjecture 3.2.1. The interior of quasi-Teichmuller space S(D∗,Γ) in B2(D

∗,Γ) is equal

to the Bers embedding T (D∗,Γ) of the Teichmuller space of a Fuchsian group Γ acting on

D∗.

Note that Zhuravlev (Zuravlev) [117] proved that T (D∗,Γ) is the connected component

of the interior of S(D∗,Γ) which contains the origin for an arbitrary Fuchsian group Γ (see

also [98]). Thus the conjecture is equivalent to connectedness of the interior of S(D∗,Γ).

3.3. The Bers projection. Let D be a hyperbolic domain in C and denote by E its

complement in C. We define the map Φ : Belt(E) → B2(D) by Φ(µ) = Sfµ|D , where fµ

is defined as in §1.1 for µ which is extended to C by setting µ = 0 on D. Recall here that

Belt(E) is the open unit ball of the complex Banach space L∞(E) with norm ‖ · ‖∞. It

is clear by definition that Φ(Belt(E)) = T (D). The map Φ : Belt(E) → T (D) is called

the (generalized) Bers projection. It is known that Φ : Belt(E) → B2(D) is holomorphic

(cf. [98]) and that the Frechet derivative d0Φ : L∞(E) → B2(D) of Φ at the origin is

described by

d0Φ[ν](z) = − 6

π

∫∫

E

ν(ζ)

(ζ − z)4dξdη (ζ = ξ + iη)

for ν ∈ L∞(E). Bers [15] strengthened Ahlfors’ theorem (Theorem 2.4.1) to the following

form.

Theorem 3.3.1. The Bers projection Φ : Belt(D) → T (D∗) is a holomorphic split

submersion, in other words, the Frechet derivative of Φ at every point exists and has a

(bounded) left inverse.

Indeed, Bers showed the above theorem for the projection Φ : Belt(D,Γ) → T (D∗,Γ)

for an arbitrary Fuchsian group Γ. In particular, T (D∗,Γ) is shown to be an open subset

of B2(D∗,Γ).

3.4. Convexity. Krushkal [57] proved that the Bers embedding T (D∗) of the universal

Teichmuller space is not starlike with respect to any point, and hence, not convex in

B2(D∗). For non-starlikeness of general Teichmuller spaces, see Krushkal [60] and Toki

[111].

In spite of the above fact, the (Bers embededing of the) Teichmuller spaces enjoy many

kinds of convexity properties. We briefly list some of them in this subsection.

The most useful is perhaps the following “disk convexity” due to Zhuravlev [117], which

is shown as an application of the Grunsky inequality. A weaker version can be proved

also by the λ-lemma (see [102]).

Theorem 3.4.1 (Zhuravlev). Let Γ be a Fuchsian group acting on D∗. Suppose that a

continuous map α : D → B2(D∗,Γ) is holomorphic in D and satisfies α(∂D) ⊂ S(D∗).


Then α(D) ⊂ S(D∗,Γ). Furthermore, if α(D)∩T (D∗) is non-empty, then α(D) ⊂ T (D∗

,Γ).

Outline of the proof. For each z ∈ D, there exists a unique Fz ∈ Σ0 such that SFz= α(z).

Let B(ℓ2) denote the complex Hilbert space consisting of bounded linear operators on ℓ2.

Then the map β : D → B(ℓ2) defined by z 7→ G[Fz] turns to be holomorphic. Then the

(generalized) maximum principle implies that

supz∈D

‖β(z)‖ = supz∈∂D

‖β(z)‖ ≤ 1

and that either ‖β(z)‖ < 1 for all z ∈ D or else ‖β(z)‖ = 1 for all z ∈ D. Theorem

1.5.1 now yields that α(D) ⊂ S(D∗). If we assume that α(z0) ∈ T (D∗) for some point

z0 ∈ D in addition, then ‖β(z0)‖ < 1 and thus ‖β(z)‖ < 1 for all z ∈ D. This means that

α(D) ⊂ T (D∗) ∩B2(D∗,Γ) = T (D∗

,Γ).

We remark that the above argument is a variant of Lehto’s principle (see [13] or [67]).

A more sophisticated application of Grunsky inequality to Teichmuller spaces can be

found in [96].

Bers and Ehrenpreis [17] proved that finite dimensional Teichmuller spaces are holo-

morphically convex. Krushkal [58] strengthened it by showing that the Teichmuller space

of an arbitrary Riemann surface R is complex hyperconvex, that is to say, there exists a

negative plurisubharmonic function u(x) on Teich(R) such that u(x) → 0 when x tends to

∞. He proved it by pointing out that the function log tanh(d(x, y)) gives the Green func-

tion on Teich(R), where d(x, y) denotes the Teichmuller distance of Teich(R). Krushkal

[59] also proved that finite dimensional Teichmuller spaces are polynomially convex.

3.5. Teichmuller distance and other natural distances (metrics). In §2, we de-

fined two kinds of distances on the universal Teichmuller space; the Teichmuller distance

and the distance induced by the Bers embedding. These distances can be defined for the

Teichmuller space of an arbitrary Riemann surface. On the other hand, since Teichmuller

spaces have complex structure, it carries natural invariant distances for holomorphic maps

(see [46] or [52] as a general reference).

Let X be a complex (Banach) manifold. The Kobayashi pseudo-distance dK(x, y) is

defined as

infN∑

j=1

dD(zj−1, zj),

where the infimum is taken over all finitely many holomorphic maps fj : D → X (j =

1, . . . , N) which satisfy fj(zj) = fj+1(zj)(1 < j < N), f1(z0) = x, and fN(zN) = y. Here,


dD(z, w) denotes the Poincare distance of D:

dD(z, w) = arctanh

∣∣∣∣w − z

1 − zw

∣∣∣∣ .

The following theorem was proved by Royden [91] for finite dimensional case and by

Gardiner (see [35] or [36]) for general case. (For a simple proof using the λ-lemma, see

[32].)

Theorem 3.5.1. The Kobayashi pseudo-distance of the Teichmuller space of a Riemann

surface is equal to the Teichmuller distance.

For other invariant metrics on Teichmuller spaces, see [75, Appendix 6].

Earle [31] proved that the Caratheodory (pseudo)distance of the Teichmuller space of

an arbitrary Fuchsian group is complete.

The Weil-Petersson metric is another important (Riemannian) metric on finite dimen-

sional Teichmuller spaces. Since the complex structure of the Teichmuller space of a

general Riemann surface is modelled on a complex Banach space which may not be re-

flexive, this metric cannot be defined on general Teichmuller spaces unless the structure

of the space is changed. However, some attempts were made to construct analogs of the

Weil-Petersson metric on the universal Teichmuller space, see [76], [77], [107], [108].

4. Pre-Schwarzian models

The Schwarzian derivative plays an important role in the definition of the Teichmuller

space. But, it is not easy to treat with Schwarzian derivative, in general, because of its

complicated form. Therefore, some attempts of replacing Schwarzian by pre-Schwarzian

have been made. See [116] and [6]. Though the pre-Schwarzian model is sometimes

called “poor man’s model” (cf. [43]) since it does not have much invariance, this model is

interesting in connection with geometric function theory.

When dealing with pre-Schwarzian derivative, the point at infinity plays a special role.

Therefore, we have to consider the case ∞ ∈ D separately.

4.1. The models T (D) and T (H). Let ∆ be a disk or a half-plane in C. Set

S(∆) = Tf : f : ∆ → C is holomorphic and univalentand

T (∆) = Tf : f : ∆ → C is holomorphic and extends to a qc map of C.

Here, Tf denotes the pre-Schwarzian derivative of f (see §1.6). By definition, T (∆) ⊂S(∆).


We recall that the pre-Schwarzian derivative vanishes only when the function is affine.

Since each circle domain in C is similar (affinely equivalent) to either the unit disk D or

the half-plane H = z ∈ C : Im z > 0. Therefore, we may restrict ourselves on the two

cases ∆ = D and H. First let f ∈ S . By the well-known inequality (cf. [29])

(4.1.1)∣∣(1 − |z|2)Tf (z) − 2z

∣∣ ≤ 4,

we obtain ‖Tf‖B1(D) ≤ 6. In particular, S(D) ⊂ B1(D). Note also that the constant 6 is

sharp as the Koebe function K(z) = z/(1−z)2 shows. It is easy to see that S(D) is closed

in B1(D).

Let L(z) = (z − i)/(z + i). Note that ‖TL‖B1(H) = 4 and hence TL ∈ T (H). Since

L∗1

: B1(D) → B1(H) is a linear isometry and TfL = L∗1(Tf ) + TL, the space T (H) is

contained in B1(H) and it is isometrically equivalent to T (D). In this sense, it is enough

to consider only T (D).

We define the map π : B1(D) → B2(D) by π(ψ) = ψ′ −ψ

2/2. By definition, π(S(D)) =

S(D) and π(T (D)) = T (D). Duren, Shapiro and Shields [30] noticed that this map is

continuous (see also §5.3).

Astala and Gehring [6] proved an analogous result to the case of Schwarzian derivative.

Theorem 4.1.2. The interior of S(D) in B1(D) is equal to T (D), while the closure of

T (D) in B1(D) is not equal to S(D). Moreover, ∂T (D) \ π(∂T (D)) is not empty.

Zhuravlev [116] revealed the following remarkable property of T (D).

Theorem 4.1.3 (Zhuravlev). The space T (D) decomposes into the uncountably many

connected components T0 and Tω, ω ∈ ∂D, where

T0 = Tf ∈ T (D) : f(D) is bounded and Tω = Tf ∈ T (D) : f(z) → ∞ as z → ω.

Moreover, ψ ∈ B1(D) : ‖ψ − ψω‖B1(D) < 1 ⊂ Tω holds for each ω ∈ ∂D, where ψω(z) =

2ω/(1 − ωz) is the pre-Schwarzian derivative of the function z/(1 − ωz).

Note that the map π is not injective even in each connected component of T (D).

Therefore, we should note that this model of the universal Teichmuller space has some

redundancy.

4.2. The model T (D∗). There is some subtlety in consideration of the pre-Schwarzian

model of the universal Teichmuller space T (D∗) on the exterior D∗ of the unit circle. The


first thing to note is the fact that the Banach space B1(D∗) is not the right space on which

T (D∗) is modeled. We define

S(D∗) = TF : F ∈ Σand

T (D∗) = TF : F ∈ Σ extends to a quasiconformal map of C.If F (ζ) = ζ + b0 + b1/ζ + b2/ζ

2 + . . . , then TF (ζ) = 2b1/ζ3 + · · · = O(ζ−3) as ζ → ∞.

Therefore, the norm

(4.2.1) B(ψ) = supζ∈D∗

(|ζ|2 − 1)|ζψ(ζ)|

is more natural. Indeed, Becker’s univalence criterion [12] and Avhadiev’s inequality [8]

(4.2.2) (|ζ|2 − 1)

∣∣∣∣ζF

′′(ζ)

F ′(ζ)

∣∣∣∣ ≤ 6

imply the following result.

Theorem 4.2.3. If a meromorphic function F (ζ) = ζ+b0 +b1/ζ+ . . . in |ζ| > 1 satisfies

B(TF ) ≤ 1, then F ∈ Σ. Conversely, every function F in Σ satisfies B(TF ) ≤ 6.

We setB

′1(D∗) = ψ ∈ B1(D

∗) : limζ→∞

ζ2ψ(ζ) = 0.

Then, it is easy to see that B′1(D∗) = ψ : D

∗ → C holomorphic and B(ψ) < ∞. The

above theorem now yields that S(D∗) is a bounded subset of B′1(D∗).

We define π : B′1(D∗) → B2(D

∗) as before by π(ψ) = ψ′ − ψ

2/2. Then π is continuous

[13, Lemma 6.1]. By definition, π(S(D∗)) = S(D∗) and π(T (D∗)) = T (D∗). Since T (D∗)

is an open set and T (D∗) = π−1(T (D∗)), the set T (D∗) is also open in B

′1(D∗). In this

way, we see that the space T (D∗) is a complex Banach manifold modeled on B′1(D∗). We

remark that π does not map B1(D∗) into B2(D

∗).

The set T (D∗) seems to be less investigated, but could be more useful. For instance,

the mapping F 7→ TF sends Σ0 to S(D∗) bijectively. Recall that the mapping F 7→ SF

sends Σ0 to S(D∗) bijectively. Therefore, the mapping π sends S(D∗) to S(D∗) bijectively.

4.3. Loci of typical subclasses of S . Since the differential operator Tf is closely

related with geometric function theory, many classical subclasses of univalent functions

correspond to sets with nice properties in S(D).

We recall several fundamental classes in univalent function theory. We denote by A the

set of analytic functions f in the unit disk D so normalized that f(0) = 0 and f ′(0) = 1.

A function f ∈ A is called starlike (convex) if f is univalent and if f(D) is starlike with


respect to the origin (convex). We denote by S ∗ and K the sets of starlike and convex

functions in A , respectively. A function f ∈ A is called close-to-convex if eiαf ′/g

′ has

positive real part in D for a convex function g and for a real constant α. Denote by C the

set of close-to-convex functions in A . It is known that C ⊂ S (cf. [29]).

It is interesting to see how pre-Schwarzians of those functions are located in the space

S(D). The following result gives an answer to this question.

Theorem 4.3.1 ([27], [51]). Tf : f ∈ K and Tf : f ∈ C are both convex subsets of

S(D).

It may be natural to conjecture the following.

Conjecture 4.3.2 ([48]). The subset Tf : f ∈ S ∗ of S(D) is starlike with respect to

the origin.

Note that the vector operations in B1(D) is translated to the Hornich operations in

the space of uniformly locally univalent functions (see, for example, [48]). Also, see Casey

[22] for relations between subclasses of S and (the closure) of T (D).

5. Univalence criteria

As is well developed in Lehto’s textbook [67], univalence criteria are closely connected

with the universal Teichmuller space. The present section will be devoted to this topic.

5.1. Univalence criteria due to Nehari and Ahlfors-Weill. Nehari [78] proved the

following result, which is fundamental in the Teichmuller spaces.

Theorem 5.1.1. Every meromorphic univalent function f on the unit disk satisfies

the inequality ‖Sf‖B2(D) ≤ 6. Conversely, if a meromorphic function f on the unit disk

satisfies the inequality ‖Sf‖B2(D) ≤ 2, then f must be univalent.

The constants 6 and 2 are sharp since the Koebe function K(z) = z/(1 − z)2 satisfies

‖SK‖B2(D) = 6 and since the function f(z) = ((1 + z)/(1− z))iǫ, ǫ > 0, is never univalent

but ‖Sf‖B2(D) = 2(1 + ε2) can approach 2 (Hille [44]). The former assertion was first

proved by Kraus [56] and reproved by Nehari. Therefore, it is called nowadays the Kraus-

Nehari theorem. The Kraus-Nehari theorem is a consequence of the Bieberbach theorem.

By the Mobius invariance of (1− |z|2)2|Sf (z)|, it is enough to show the inequality only at

the origin, namely, |Sf (0)| ≤ 6 for f ∈ S . A straightforward computation gives Sf (0) =

6(a3−a2

2) for f(z) = z+a2z

2 +a3z3 + . . . . If we set F (ζ) = 1/f(1/ζ) = ζ+b0 +b1/ζ+ . . . ,

then b1 = a2

2− a3, and thus the inequality |b1| ≤ 1 (see (1.5.3)) implies the required one.


The class N = f ∈ A : ‖Sf‖B2(D) ≤ 2 is sometimes called the Nehari class. Gehring

and Pommerenke [41] showed that f ∈ N maps the unit disk conformally onto a Jordan

domain unless f(D) is Mobius equivalent to the parallel strip z : |Im z| < π/4. For

further development, see [24], [25] and [26].

In connection with Nehari’s theorem, Ahlfors and Weill established the following quasi-

conformal extension criterion. For ϕ ∈ B2(D∗) with ‖ϕ‖B2(D∗) < 2, we set α[ϕ] ∈ Belt(D)

by α[ϕ](z) = −ρD(z)−2ϕ(1/z)z−4

/2. Note that the map α is the restriction of a bounded

linear operator which maps B2(D∗,Γ)2 = ϕ ∈ B2(D

∗,Γ) : ‖ϕ‖B2(D) < 2 into Belt(D∗

,Γ)

for every Fuchsian group Γ.

Theorem 5.1.2 (Ahlfors-Weill). The map α : B2(D∗)2 → Belt(D) is the local inverse

of the Bers projection Φ : Belt(D) → T (D∗), in other words, Φ(α[ϕ]) = ϕ for ϕ ∈ B2(D∗)

with ‖ϕ‖B2(D∗) < 2.

Corollary 5.1.3. The universal Teichmuller space T(D∗) contains the open ball centered

at the origin with radius 2 in B2(D∗).

The map α : B2(D∗)2 → Belt(D) is sometimes called the Ahlfors-Weill section.

5.2. Inner radius and outer radius. Let D be a hyperbolic domain in C. The inner

radius σI(D) and the outer radius σO(D) of univalence is defined respectively by

σI(D) = supσ ≥ 0 : ‖Sf‖B2(D) ≤ σ ⇒ f is univalent in D,

σO(D) = sup‖Sf‖B2(D) : f : D → C is univalent.

We also define the number τ(D) ∈ [0,+∞] as ‖Sp‖B2(D), where p is a holomorphic universal

covering projection of D onto D. The quantity τ(D) is independent of the choice of p and

thus well defined. Note that τ(D) <∞ if and only if ∂D is uniformly perfect (cf. [87] or

[103]).

Summarizing theorems of Ahlfors [2], Gehring [38], Nehari [78], we obtain the following.

Theorem 5.2.1. σI(∆) = 2, σO(∆) = 6, τ(∆) = 0 hold for a circle domain ∆. Let D be

a simply connected hyperbolic domain. Then σO(∆) ≤ 12 and τ(D) ≤ 6. Moreover, D is

a quasidisk if and only if σI(D) > 0.

The inequality σO(∆) ≤ 12 is shown as follows. Let f : D → C be univalent and set

Ω = f(D). Take a conformal map g : D∗ → D and set h = f g. Then, by Lemmas 1.3.1,

1.6.2 and the Kraus-Nehari theorem, we obtain

‖Sf‖B2(D) = ‖g∗2(Sf )‖B2(D∗) = ‖Sh − Sg‖B2(D∗) ≤ ‖Sh‖B2(D∗) + ‖Sg‖B2(D∗) ≤ 12.


It is a remarkable fact due to Beardon and Gehring [10] that σO(D) ≤ 12 holds even for

an arbitrary hyperbolic domain D.

The inner and outer radii of univalence are better understood in the context of (quasi-)

Teichmuller space.

Lemma 5.2.2. Let g : D∗ → D be a conformal homeomorphism of D

∗ onto a simply

connected hyperbolic domain D. Then ϕ ∈ S(D∗) : ‖ϕ−Sg‖ < σI(D) is the maximal open

ball centered at Sg contained in T (D∗). On the other hand, σO(D) = max‖ϕ−Sg‖B2(D∗) :

ϕ ∈ S(D∗).

Lehto [65] proved the following relations.

Theorem 5.2.3. The relation σO(D) = τ(D) + 6 holds for a simply connected hyperbolic

domain D. Furthermore, 2 − τ(D) ≤ σI(R) ≤ min2, 6 − τ(D).

As for the quantity τ(D), the following are known. For a convex domain D, we have

τ(D) ≤ 2. This result is repeatedly re-discovered by many mathematicians; [85], [90],

[112], [79], [65]. Suita [106] refined this result by showing the sharp inequality

τ(f(D)) ≤

2, 0 ≤ α ≤ 1/2,

8α(1 − α), 1/2 ≤ α ≤ 1

for a convex function f ∈ K of order α, namely, when Re (1 + zf′′(z)/f ′(z)) > α.

It is known that τ(D) ≤ 6(K2 − 1)/(K2 + 1) for a K-quasidisk D (see [67]). See also

[68], [53], [23], [72], [9] for other classes of domains.

It is not easy to determine, or even to estimate from below, the value of σI(D), in

general. Known examples are sectors [62], triangles [64], the interiors and the exteriors

of regular polygons [21], [64], some other polygonal domains [73], [74], the exteriors of

hyperbolas [63].

For a general method of estimating σI(D) from below, see [66], [67] and [104]. See also

[105].

5.3. Pre-Schwarzian counterpart. One can define quantities similarly as in the pre-

vious section with respect to pre-Schwarzian derivative. We add the symbol ˆ to indicate

it. For instance,

σI(D) = supσ ≥ 0 : ‖Tf‖B1(D) ≤ σ ⇒ f is univalent in D

for a hyperbolic domain D in C. In the case when D = D∗, we adopt the norm B(ψ) :

σI(D∗) = supσ ≥ 0 : B(TF ) ≤ σ ⇒ f is univalent in D

∗.


Duren, Shapiro and Shields [30] proved that σI(D) ≥ 2(√

5−2) = 0.472 · · · by observing

that ‖ψ′‖B2(D) ≤ 4‖ψ‖B1(D) and thus π(ψ) = ψ′ −ψ

2/2 is a continuous map of B1(D) into

B2(D). Note that Wirths [114] found the sharp constant C = (13√

3 + 55√

11)/64 =

3.20204 . . . for the estimate ‖ψ′‖B2(D) ≤ C‖ψ‖B1(D). Nowadays, the best value for this

univalence criterion is known.

Theorem 5.3.1. σI(∆) = 1 and σO(∆) = 6 for ∆ = D,H and D∗.

Becker [11], [12] showed that σI(D) ≥ 1 and σI(D∗) ≥ 1 and Becker-Pommerenke [14]

showed that equality hold for ∆ = D and that σI(H) = 1. Pommerenke [88] showed the

sharpness for ∆ = D∗.

By (4.1.1) and the fact that the Koebe function K satisfies ‖TK‖B1(D) = 6, we see that

σO(D) = 6. σO(H) = 6 can be seen by noting the relation

‖ψ‖B1(H) = limr→1−

‖ψ‖B1(∆r)

for ψ ∈ B1(H), where ∆r = z : |z − i(1 + r2)/(1 − r

2)| < 2r/(1 − r2). The formula

σO(D∗) = 6 follows from the fact that the inequality in (4.2.2) is sharp for each ζ.

For concrete estimates of τ(D) for several geometric classes of domains, see [115], [101],

[81], [49], [50].

In spite of relative simplicity of the operation Tf , very little is known for quantities

σI(D) and σO(D). Stowe [97] gave non-trivial examples of domains D for which σI(D) ≥ 1.

5.4. Directions of further investigation. The Bers embedding of Teichmuller spaces

is still mysterious. We know very little about the shape of it. Pictures of one-dimensional

Teichmuller spaces were recently given in [54] and [55]. Note that the first attempt towards

it was done by Porter [89] as early as in 1970’s.

It is an interesting and important problem to describe the intersection of T (∆) or T (∆)

with a (complex) one-dimensional vector subspace of B2(∆) or B1(∆) for a circle domain.

Completely known examples are essentially, as far as the author knows, the linear hull of

1/(1 − z) in B1(∆) [92] and the linear hull of z−2 in B2(H) in [47], only.

The results presented above could be generalized to various directions. We end this

survey with remarks on possible ways to study furthermore.

In this section, we considered mainly the case when the domain is simply connected.

When the domain is multiply connected, the problem will become much more difficult.

See [83] and [84] for fundamental information.

We were concerned here with only pre-Schwarzian and Schwarzian derivatives. On the

other hand, several definitions of higher-order Schwarzian derivatives have been proposed

(e.g., [109], [93]). Thus, we may develop the theory for those higher-order Schwarzian

derivatives.


Of course, we may consider domains in Cn or R

n but with great difficulty caused by

the lack of canonical metrics such as hyperbolic metric, the lack of Riemann mapping

theorem and so on. Note that Martio and Sarvas [70] gave some injectivity conditions

even in higher dimensions.

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Toshiyuki Sugawa Address: Department of Mathematics, Graduate School of Science, Hiroshima

University, Higashi-Hiroshima, 739-8526 Japan



Metrics and quasiregular mappings

Matti Vuorinen

Abstract. This series of lectures intends to provide a gateway to some se-lected topics of quasiconformal and quasiregular maps, in particular to themain themes of [Vu1] and [Vu3]. Some of the basic notions and tools arebriefly reviewed. Several problems, exercises and open problems are giventhroughout the text. At the end of the paper a short list of some genericopen problems is presented for metric spaces, which allow a great number ofvariations in specific cases.

Keywords. quasiregular mappings, metric spaces.

2000 MSC. 30C65.

Contents

1. Introduction 291

2. Mobius transformations 295

3. Hyperbolic geometry 300

4. Quasihyperbolic geometry 305

5. Modulus and capacity 306

6. Conformal invariants 313

7. Distortion theory 319

8. Open problems 322

References 323

1. Introduction

The goal of these lectures is to provide an introduction to some of the mainproperties of quasiconformal and quasiregular mappings. One of the centralthemes here will be to study how these mappings deform distances and metricsand therefore it is natural to study our mappings between metric spaces. In mostcases, the metrics will have some useful invariance or quasi-invariance propertiesunder a set Γ of transformations, called rigid motions. An important example isthe unit ball of R

n equipped with the hyperbolic metric in which case we maytake the set Γ to be the group of the Mobius self-mappings of the ball.

Version October 19, 2006.

292 M. Vuorinen IWQCMA05

The material is largely drawn from [Vu3] and [AVV2]. In order to give thereader a chance to enter gradually this territory of mathematical research, prob-lems of varying level are given, from easy exercises to research problems. Manymore can be found in [Vu3] and [AVV2] (the exercises in [AVV2] come with so-lutions). Some research problems are collected at the end of the paper. Becauseof limitations of space, most of the details/proofs are omitted with the generalreference to [V1] and [Vu3].

The idea of using invariance with respect to rigid motion to study functiontheory is very old. In fact, it can be traced back to nineteenth century, inparticular, to the work of F. Klein. Perhaps the most natural notion of invarianceis conformal invariance under the group of conformal self-maps of a given simply-connected domain. Several conformal invariants emerged from the studies of H.Grotzsch, L. Ahlfors, and A. Beurling.

A pair (X, d) is called a metric space if X 6= ∅ and d : X×X → [0,∞) satisfiesthe following four conditions

(1.1)

(M1) d(x, y) ≥ 0 for all x, y ∈ X ,

(M2) d(x, y) = 0 iff x = y ,

(M3) d(x, y) = d(y, x) for all x, y ∈ X ,

(M4) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X .

Let (X, d1) and (Y, d2) be metric spaces and let f : X → Y be a continuousmapping. Then we say that f is uniformly continuous if there exists an increasingcontinuous function ω : [0,∞) → [0,∞) with ω(0) = 0 and d2(f(x), f(y)) ≤ω(d1(x, y)) for all x, y ∈ X . We call the function ω the modulus of continuity off . If there exist C,α > 0 such that ω(t) ≤ Ct

α for all t > 0 , we say that f isHolder-continuous with Holder exponent α . If α = 1 , we say that f is Lipschitzwith the Lipschitz constant C or simply C-Lipschitz. If f is a homeomorphismand both f and f

−1 are C-Lipschitz, then f is C-bilipschitz or C-quasiisometryand if C = 1 we say that f is an isometry. These conditions are said to holdlocally, if they hold for each compact subset of X .

1.2. Exercise. If h : [0,∞) → [0,∞) is a function and h(t)/t is decreasing,show that h(x + y) ≤ h(x) + h(y) for all x, y > 0. In particular, show that if(X, d) is a metric space, then also (X, d

α), α ∈ (0, 1), is.

1.3. Exercise. Let f : [0,∞) → [0,∞) be Holder-continuous with exponentβ > 1. Show that f(x) = f(0) for all x > 0 .

1.4. Example. Let f : Rn → R

n, f(x) = |x|α−1

x, f(0) = 0. Then f is Holder-continuous with exponent α .

In most examples below, the metric spaces will have some additional structure.The metrics will often have some quasiinvariance properties. For instance, wesay that a pair of metric spaces (Xj, dj), j = 1, 2, is quasiinvariant under a setΓ of mappings f : (X1, d1) → (X2, d2) if there exists C ≥ 1 such that 1/C ≤d2(f(x), f(y))/d1(x, y) ≤ C for all x, y ∈ X1, x 6= y and all f ∈ Γ . In particular,

Metrics and quasiregular mappings 293

we will study metric spaces (X, d) where the group Γ of automorphisms of X

acts transitively (i.e. given x, y ∈ X there exists h ∈ Γ such that hx = y .) IfC = 1, then we say that d is invariant.

1.5. Examples. (1) The euclidean space Rn equipped with the usual metric

|x − y| = (∑n

j=1(xj − yj)

2)1/2, Γ is the group of translation in R

n.

(2) The unit sphere Sn = z ∈ R

n+1 : |z| = 1 equipped with the metric ofR

n+1 and Γ is the set of rotations of Sn.

(3) Let G ⊂ Rn, G 6= R

n, for x, y ∈ G set

jG(x, y) = log

(1 +

|x − y|mind(x, ∂G), d(y, ∂G)

).

Then one can prove that jG is a metric (this is a folklore result, see e.g. [S]). Infact, there exists a constant C > 1 such that for the unit ball Bn of R

n

1/C ≤ ρBn(x, y)/jBn(x, y) ≤ C

for all x, y ∈ Bn, x 6= y . Here ρBn is the hyperbolic metric of Bn and it is

invariant under the group of Mobius self-mappings of Bn. For the definition of

ρBn see below or [Vu3, Section2].

A basic geometric object of a metric space (X, d) is the ball BX(z, r) = x ∈X : d(x, z) < r . In order to study how balls and their boundary spheres aredeformed under homeomorphisms, we introduce a deformation measure Hf (x0, r)of a ball under a homeomorphism f : (X1, d1) → (X2, d2) at a point x ∈ X1

Hf (x, r) = sup

d2(f(x), f(y))

d2(f(x), f(z)): d1(z, x) = d1(y, x) = r

.

x

rr

f

lr f(x)

L

Figure 1. Hf (x, r) .

If f maps spheres centered at x onto spheres centered at f(x) , then Hf (x, r) =1. For instance the above radial mapping x 7→ |x|α−1

x has the property Hf (0, r) =1 for all r > 0 . Recall from Complex Analysis that a conformal map f : D1 →D2, Dj ⊂ C, j = 1, 2, satisfies limr→0 Hf (x, r) = 1 for all x ∈ D1 . We say that


a homeomorphism f : (X1, d1) → (X2, d2) is quasiconformal (with respect to(d1, d2)), if there exists C ∈ [1,∞) such that for all x ∈ X1

Hf (x) = lim supr→0

Hf (x, r) ≤ C .

If f is L-bilipschitz, then f satisfies the above condition with C = L2 .

Let Gj ⊂ Rn, j = 1, 2, be domains and let f : G1 → G2 be a homeomorphism.

Suppose now that there exists a constant C ≥ 1 such that for all subdomainsD ⊂ G1 the mapping f |D : (D, jD) → (f(D), jf(D)) is C-Lipschitz. Fix x0 ∈ G1

and r ∈ (0, d(x0, ∂G1)/2) . If |x − x0| = |y − x0| = r and G = Bn(x0, 2r) \ x0 ,

then jG(x, y) ≤ log 3 and we obtain by the above C-Lipschitz-property∣∣∣∣log

|f(x) − f(x0)||f(y) − f(x0)|

∣∣∣∣ ≤ jfG(f(x), f(y)) ≤ CjG(x, y) ≤ C log 3 ,

and hence Hf (x0) ≤ 3C, where we used the triangle inequality (Lemma 3.21 (3)

below) and the fact that x0 ∈ ∂G . Now d1 and d2 are the usual metrics. Thuswe see that our map is quasiconformal.

In this argument the fact that x0 ∈ ∂G played a key role. For most ofthe metrics that we will consider here, even one single boundary point will beimportant. Most of the metrics will also be monotone with respect to the domain.Thus, for instance, if G1 ⊂ G2 ⊂ R

n are domains, then jG1(x, y) ≥ jG2

(x, y) forall x, y ∈ G1 and for a fixed x0 ∈ G1 , jG1

(x0, x) → ∞ as x → x1 ∈ ∂G . Inthe above argument, it was assumed that f |D is Lipschitz continuous for allsubdomains D of G1 but we only used this property for subdomains of the formB

n(x, r) \ x , x ∈ G1 . In order to motivate this condition let us recall that aconformal map is conformal also in every subdomain.

Here we have studied the metric j , mainly because it is very easy to defineand because it well represents the metrics we study here. There are now severalquestions:

(a) Can we characterize the class of quasiisometries or isometries in the abovesense?

(b) Can we prove similar results for other metrics (and what are these met-rics)?

(c) Can we say more for the case when the domains are ”nice” (for instancequasidisks)?

Conformal invariants and conformally invariant metrics have been an impor-tant topic in geometric function theory during the past century. One of the firstpromoters of these ideas was F. Klein. In the context of quasiconformal map-pings these ideas emerged as a result of the pioneering works of H. Grotzsch, O.Teichmuller, L. Ahlfors and A. Beurling on quasiconformal maps in plane do-mains [LV], [K]. Extension to higher dimensions is due to F.W. Gehring and J.Vaisala [G] , [V1]. The case of metric measure spaces has been studied recentlyby J. Heinonen, P. Koskela and many other people [H].


In the setup presented here, the aforementioned questions (a)-(c) were studiedalready in [Vu1] and [Vu3]. But only very few answers are known, see [H1], [H2],[HI]. These questions could also be investigated for some particular classes of do-mains, which would bring a very wide spectrum of new questions into play. Someexamples of such classes of domains would be uniform domains and quasiconfor-mal balls. As we will see, there still are numerous open problems in this area.It is assumed that the reader is familiar with some basic facts and definitions ofthe theory of quasiconformal and quasiregular maps [V1], [Vu3].

2. Mobius transformations

For x ∈ Rn and r > 0 let

Bn(x, r) = z ∈ R

n : |x − z| < r ,S

n−1(x, r) = z ∈ Rn : |x − z| = r

denote the ball and sphere, respectively, centered at x with radius r. The abbre-viations B

n(r) = Bn(0, r), S

n−1(r) = Sn−1(0, r), Bn = B

n(1), Sn−1 = S

n−1(1)will be used frequently. For t ∈ R and a ∈ R

n \ 0 we denote

P (a, t) = x ∈ Rn : x · a = t ∪ ∞.

Then P (a, t) is a hyperplane in Rn

= Rn ∪ ∞ perpendicular to the vector a,

at distance t/|a| from the origin.

2.1. Definition. Let D and D′ be domains in R

n and let f : D → D′ be a

homeomorphism. We call f conformal if (1) f ∈ C1, (2) Jf (x) 6= 0 for allx ∈ D, and (3) |f ′(x)h| = |f ′(x)||h| for all x ∈ D and all h ∈ R

n. If D andD

′ are domains in Rn, we call a homeomorphism f : D → D

′ conformal if therestriction of f to D \ ∞, f

−1(∞) is conformal.

2.2. Examples. Some basic examples of conformal mappings are the followingelementary transformations.

(1) A reflection in P (a, t):

f1(x) = x − 2(x · a − t)a

|a|2 , f1(∞) = ∞ .

(2) An inversion (reflection) in Sn−1(a, r):

f2(x) = a +r2(x − a)

|x − a|2 , f2(a) = ∞ , f2(∞) = a .

(3) A translation f3(x) = x + a , a ∈ Rn, f3(∞) = ∞.

(4) A stretching by a factor k > 0: f4(x) = kx , f4(∞) = ∞.

(5) An orthogonal mapping, i.e. a linear map f5 with

|f5(x)| = |x| , f5(∞) = ∞ .


2.3. Remarks. (1) The translation x 7→ x + a can be written as a compositionof reflections in P (a, 0) and P (a,

1

2|a|2). The stretching x 7→ kx, k > 0, can be

written as a composition of inversions in Sn−1(0, 1) and S

n−1(0,√

k ). It can beproved, that an orthogonal mapping can be composed of at most n+1 reflectionsin planes (see [BE, p. 23, Theorem 3.1.3]).

(2) It is easy to show that f1(f1(x)) = x and f2(f2(x)) = x for all x ∈ Rn, i.e. f1

and f2 are involutions.

(3) It can also be shown that we have the difference formula

|f2(x) − f2(y)| =r2|x − y|

|x − a||y − a|for all x, y ∈ R

n \ a.(4) If an = 0, then one can show that f2(H

n) = Hn and that for all x, y ∈ Hn

|f2(x) − f2(y)|2(f2(x))n(f2(y))n

=|x − y|2(x)n(y)n

.

(5) Reflections and inversions are sense-reversing. The composition of two sense-reversing maps is sense-preserving.

2.4. Definition. A homeomorphism f : Rn → R

nis called a Mobius transfor-

mation if f = g1 · · · gp where each gj is one of the elementary transformationsin 2.2(1)–(5) and p is a positive integer. Equivalently (see 2.3) f is a Mobiustransformation if f = h1 · · · hm where each hj is a reflection in a sphere or in a

hyperplane and m is a positive integer. If G ⊂ Rn

the set of all (sense-preserving)Mobius transformations mapping G onto itself is denoted by GM(G) (M(G)).

It will be convenient to identify Rn

with the subset x ∈ Rn : xn+1 = 0 ∪∞

of Rn+1

. The identification is given by the embedding

x 7→ ˜x = (x1, . . . , xn, 0) ; x = (x1, . . . , xn) ∈ Rn

.(2.5)

We are now going to describe a natural two–step way of extending a Mobiustransformation of R

n to a Mobius transformation of Rn+1. First, if f in GM(R

n)

is a reflection in P (a, t) or in Sn−1(a, r), let ˜

f be a reflection in P (˜a, t) or Sn(˜a, r),

respectively. Then if x ∈ Rn

and y = f(x), by 2.2(1)–(2) we get

˜f(x1, . . . , xn, 0) = (y1, . . . , yn, 0) =

˜f(x) .(2.6)

By (2.6) we may regard ˜f as an extension of f . Note that ˜

f preserves the planexn+1 = 0 and each of the half–spaces xn+1 > 0 and xn+1 < 0. These facts followfrom the formulae 2.2(1)–(2). Second, if f is an arbitrary mapping in GM(R

n)

it has a representation f = f1 · · · fm where each fj is a reflection in a plane

or a sphere. Then ˜f = ˜

f1 · · · ˜fm is the extension of f , and it preserves the

half–spaces xn+1 > 0, xn+1 < 0, and the plane xn+1 = 0. In conclusion, every

f in GM(Rn) has an extension ˜

f in GM(Rn+1

). It follows from [BE, p. 31,


Theorem 3.2.4] that such an extension ˜f of f is unique. The mapping ˜

f is called

the Poincare extension of f . In the sequel we shall write x, f instead of ˜x,˜f ,

respectively.

Many properties of plane Mobius transformations hold for n–dimensionalMobius transformations as well. The fundamental property that spheres of R

n

(which are spheres or planes in Rn, see [Vu1, Exercise 1.26, p.8]) are preserved

under Mobius transformations is proved in [BE, p. 28, Theorem 3.2.1].

2.7. Stereographic projection. The stereographic projection π : Rn →

Sn(1

2en+1,

1

2) is defined by

π(x) = en+1 +x − en+1

|x − en+1|2, x ∈ R

n ; π(∞) = en+1(2.8)

Then π is the restriction to Rn

of the inversion in Sn(en+1, 1). In fact, we can

identify π with this inversion. Because f−1 = f for every inversion f , it follows

that π maps the “Riemann sphere” Sn(1

2en+1,

1

2) onto R

n.

The spherical (chordal) metric q in Rn

is defined by

q(x, y) = |π(x) − π(y)| ; x, y ∈ Rn

,(2.9)

where π is the stereographic projection (2.8). From the definition (2.8) by cal-culating we obtain

q(x, y) =|x − y|√

1 + |x|2√

1 + |y|2; x 6= ∞ 6= y,

q(x,∞) =1√

1 + |x|2.

(2.10)

For x ∈ Rn \ 0 the antipodal (diametrically opposite) point x, is defined by

x = − x

|x|2(2.11)

and we set ∞ = 0, 0 = ∞ . Then, by (2.10), q(x, x ) = 1 and hence π(x),π(x )are indeed diametrically opposite points on the Riemann sphere.

2.12. Balls in the spherical metric. For x ∈ Rn

and r ∈ (0, 1) we define thespherical ball

Q(x, r) = z ∈ Rn

: q(x, z) < r .(2.13)

Its boundary sphere is denoted by ∂Q(x, r). From the Pythagorean theorem itfollows that (cf. (2.11))

Q(x, r) = Rn \ Q

(x,

√1 − r2

).(2.14)


x

y

_n

0

π(x)

π(y)

en+1

Figure 2. Formulae (2.8) and (2.9) visualized.

Figure 3. A cross-section of the Riemann sphere.

To gain insight into the geometry of spherical balls Q(x, r) it is convenientto study the image πQ(x, r) under the stereographic projection π (see figure 3).Indeed, by definition (2.9) we see that

πQ(x, r) = Bn+1(π(x), r

)∩ S

n(1

2en+1,

1

2).(2.15)

Either by this formula or more directly by the definition of the spherical metric(plus the fact that Mobius transformations preserve spheres) we see that in theeuclidean geometry, Q(x, r) is a point set of one of the following three kinds

(a) an open ball Bn(u, s),

(b) the complement of Bn(v, t) in R

n,(c) a half–space of R

n.


Clearly, ∂Q(x, r) is either a sphere or a hyperplane of Rn. Formula (2.14)

shows, in particular, that πQ(x, 1/√

2 ) is a half–sphere of the Riemann sphereS

n(1

2en+1,

1

2).

2.16. Absolute ratio. For an ordered quadruple a, b, c, d of distinct points inR

nwe define the absolute (cross) ratio by

| a, b, c, d | =q(a, c) q(b, d)

q(a, b) q(c, d).(2.17)

It follows from (2.10) that for distinct a, b, c, d in Rn

| a, b, c, d | =|a − c| |b − d||a − b| |c − d| .

One of the most important properties of Mobius transformations is that theypreserve absolute ratios, i.e. if f ∈ GM, then

| f(a), f(b), f(c), f(d) | = | a, b, c, d |(2.18)

for all distinct a, b, c, d in Rn. As a matter of fact, the preservation of absolute

ratios is a characteristic property of Mobius transformations. It is proved in [BE,p. 72, Theorem 3.2.7] that a mapping f : R

n → Rn

is a Mobius transformationif and only if f preserves all absolute ratios.

2.19. Automorphisms in Bn. We shall give a canonical representation forthe maps in M(Bn). Assume that f is in M(Bn) and that f(a) = 0 for somea ∈ Bn. We denote

a∗ =

a

|a|2 , a ∈ Rn \ 0(2.20)

and 0∗ = ∞, ∞∗ = 0. Fix a ∈ Bn \ 0. Let

σa(x) = a∗ + r

2(x − a∗)∗ , r

2 = |a|−2 − 1(2.21)

be an inversion in the sphere Sn−1(a∗

, r) orthogonal to Sn−1. Then σa(a) = 0,

σa(a∗) = ∞, σa(B

n) = Bn.

Let pa denote the reflection in the (n − 1)–dimensional plane P (a, 0) through theorigin and orthogonal to a and define a sense–preserving Mobius transformationby Ta = pa σa. Then, by (2.21), TaB

n = Bn, Ta(a) = 0, and with ea = a/|a| wehave Ta(ea) = ea, Ta(−ea) = −ea. For a = 0 we set T0 = id, where id stands forthe identity map. The proof of the following fundamental fact can be found in[A, p. 21], [BE, p. 40, Theorem 3.5.1].

We now define a spherical isometry tz in M(Rn) which maps a given pointz ∈ R

n to 0 as follows. For z = 0 let tz = id and for z = ∞ let tz = p f , wheref is inversion in S

n−1 and p is reflection in the (n−1)–dimensional plane x1 = 0.

For z ∈ Rn \ 0 let sz be inversion in S

n−1(−z/|z|2, r), where r =√

1 + |z|−2.According to [Vu1, (1.45)], the inversion sz is a spherical isometry and it is easyto show that sz(z) = 0. Let pz be reflection in the plane P (z, 0). Defining

tz = pz sz ,(2.22)


0 a b

r1

Ss

Figure 4. Inversion in Sn−1

, b = a∗.

we see that tz ∈ M(Rn) is a spherical isometry with tz(z) = 0. Hence

(2.23)

tz(Q(z, r)) = Q(0, r) = Bn(r/

√1 − r2

),

|tz(x)|2 =q(x, z)2

1 − q(x, z)2

for all x, z ∈ Rn, r ∈ (0, 1).

2.24. Lemma. Let a ∈ Rn, r > 0, and let b ∈ R

n, u > 0, be such that

Bn(a, r) = Q(b, u). If f is the inversion in S

n−1(a, r), then

f = t−1

b f1 tb ,

where tb is the spherical isometry defined in (2.22) and f1 is the inversion in

Sn−1(u/

√1 − u2 ) = ∂Q(0, u).

3. Hyperbolic geometry

Hyperbolic geometry can be developed in the context of two spaces or, as theyare sometimes called, models. These two models of the hyperbolic space are theunit ball Bn and the Poincare half–space

Hn = Rn+

= (x1, . . . , xn) ∈ Rn : xn > 0 .

These two models can be equipped with a hyperbolic metric ρ that is uniqueup to a multiplicative constant in either model. In either model the metric isnormalized (by giving the element of length of the metric) in such a way that forall x, y ∈ Bn

ρHn

(h(x), h(y)

)= ρBn(x, y)

whenever h ∈ GM and hBn = Hn. Therefore both models are conformallycompatible in the sense that the two metric spaces (Bn

, ρ) and (Hn, ρ) can be

identified. This compatibility is very convenient in computations because we may


do a computation in that model in which it is easier, without loss of generality.In what follows we shall use the symbols R

n+

and Hn interchangeably.

For A ⊂ Rn let A+ = x ∈ A : xn > 0 . We define a weight function

w : Rn+→ R+ = x ∈ R : x > 0 by

w(x) =1

xn

, x = (x1, . . . , xn) ∈ Rn+

.(3.1)

If γ : [0, 1) → Rn+

is a continuous mapping such that γ[0, 1) is a rectifiable curvewith length s = ℓ(γ), then γ has a normal representation γ

0 : [0, s) → Rn+

parametrized by arc length (see J. Vaisala [V, p. 5]). The hyperbolic length

of γ[0, 1) is defined by

ℓh(γ[0, 1)) =

∫ s

0

|(γ0)′(t)| w(γ0(t))dt =

∫

γ

|dx|xn

.(3.2)

If A ⊂ Rn+

is a (Lebesgue) measurable set we define the hyperbolic volume of A

by

mh(A) =

∫

A

w(x)ndm(x) ,(3.3)

where m stands for the n–dimensional Lebesgue measure and w is as in (3.1). Ifa, b ∈ R

n+, then the hyperbolic distance between a and b is defined by

ρ(a, b) = infα∈Γab

ℓh(α) = infα∈Γab

∫

α

|dx|xn

,(3.4)

where Γab stands for the collection of all rectifiable curves in Rn+

joining a and b.Sometimes the more complete notation ρR

n

+(a, b) or ρHn(a, b) will be employed.

Figure 5. Some geodesics of Hn = Rn+

.

The infimum in (3.4) is in fact attained: for given a, b ∈ Rn+

there exists acircular arc L perpendicular to ∂R

n+

such that the closed subarc J [a, b] of L with


end points a and b satisfies

ρ(a, b) = ℓh(J [a, b]) =

∫

J [a,b]

|dx|xn

.(3.5)

If a and b are located on a normal of ∂Rn+, then J [a, b] = [a, b] = (1−t)a+tb :

0 ≤ t ≤ 1 (cf. [BE, p. 134]). Because of the (hyperbolic) length–minimizingproperty (3.5), the arc J [a, b] will be called the geodesic segment joining a and b.

Knowing the geodesics, we calculate the hyperbolic distance in two specialcases. First, for r, s > 0 we obtain

ρ(ren, sen) =∣∣∣∫ r

s

dt

t

∣∣∣ =∣∣∣ log

r

s

∣∣∣ .(3.6)

Second, if ϕ ∈ (0, 1

2π) we denote uϕ = (cos ϕ)e1 + (sin ϕ)en and calculate

ρ(en, uϕ) =

∫

J [uϕ,en]

dα

sin α=

π/2∫

ϕ

dα

sin α= log cot 1

2ϕ .(3.7)

Figure 6. The points uϕ and en .

We shall often make use of the hyperbolic functions sh x = sinh x, ch x =cosh x, th x = tanh x, cth x = coth x and their inverse functions. The aboveformulae (3.6) and (3.7) are special cases of the general formula (see [BE, p. 35])

ch ρ(x, y) = 1 +|x − y|22xnyn

, x, y ∈ Hn = Rn+

.(3.8)

Note that by this formula the hyperbolic distance ρ(x, y) is completely deter-mined once the euclidean distances xn = d(x, ∂Hn), yn = d(y, ∂Hn), and |x− y|are known. In passing we note that if f2 ∈ GM(Hn) is as defined in Remark2.3(4), then ρ(x, y) = ρ(f2(x), f2(y)) for all x, y ∈ Hn

. For another formulationof (3.8) let z, w ∈ Hn, let L be an arc of a circle perpendicular to ∂Hn withz, w ∈ L and let z∗, w∗ = L∩∂Hn, the points being labelled so that z∗, z, w, w∗occur in this order on L. Then (cf. [BE, p. 133, (7.26)])

ρ(z, w) = log | z∗, z, w, w∗ | .(3.9)


Figure 7. The quadruple z∗, z, w, w∗ .

Note that (3.6) is a special case of (3.9) when z∗ = 0 and w∗ = ∞ because| 0, z, w,∞| = |w|/|z| for z, w ∈ Hn. The invariance of ρ is apparent by (3.9)and (2.18): If f in GM(Hn), then for all x, y ∈ Hn

ρ(x, y) = ρ(f(x), f(y)) .(3.10)

For a ∈ Hn and M > 0 the hyperbolic ball x ∈ Hn : ρ(a, x) < M isdenoted by D(a,M). It is well known that D(a,M) = B

n(z, r) for some z andr (this also follows from (3.10)! ). This fact together with the observation thatλten, (t/λ)en ∈ ∂D(ten,M), λ = e

M (cf. (3.6)), yields

(3.11)

D(ten,M) = Bn((t ch M)en, t sh M

),

Bn(ten, rt) ⊂ D(ten,M) ⊂ B

n(ten, Rt) ,

r = 1 − e−M

, R = eM − 1 .

Figure 8. The hyperbolic ball D(ten,M) as a euclidean ball.

A counterpart of (3.8) for Bn is

sh2(

1

2ρ(x, y)

)=

|x − y|2(1 − |x|2)(1 − |y|2) , x, y ∈ Bn

,(3.12)

(cf. [BE, p. 40]). As in the case of Hn, we see by (3.12) that the hyperbolicdistance ρ(x, y) between x and y is completely determined by the euclidean


quantities |x − y|, d(x, ∂Bn), d(y, ∂Bn). Finally, we have also

ρ(x, y) = log |x∗, x, y, y∗| ,(3.13)

where x∗, y∗ are defined as in (3.9): If L is the circle orthogonal to Sn−1 with

x, y ∈ L, then x∗, y∗ = L∩ Sn−1, the points being labelled so that x∗, x, y, y∗

occur in this order on L. It follows from (3.13) and (2.18) that

ρ(x, y) = ρ(h(x), h(y))(3.14)

for all x, y ∈ Bn whenever h is in GM(Bn). Finally, in view of (2.18), (3.9), and(3.13) we have

ρBn(x, y) = ρHn(g(x), g(y)) , x, y ∈ Bn,(3.15)

whenever g is a Mobius transformation with gBn = Hn.

It is well known that the balls D(z,M) of (Bn, ρ) are balls in the euclidean

geometry as well, i.e. D(z,M) = Bn(y, r) for some y ∈ Bn and r > 0. Making

use of this fact, we shall find y and r. Let Lz be a euclidean line through 0 and z

and z1, z2 = Lz ∩ ∂D(z,M), |z1| ≤ |z2|. We may assume that z 6= 0 since withobvious changes the following argument works for z = 0 as well. Let e = z/|z|and z1 = se, z2 = ue, u ∈ (0, 1), s ∈ (−u, u). Then it follows that

ρ(z1, z) = log(1 + |z|

1 − |z| ·1 − s

1 + s

)= M ,

ρ(z2, z) = log(1 + u

1 − u· 1 − |z|1 + |z|

)= M .

Solving these for s and u and using the fact that

D(z,M) = Bn(

1

2(z1 + z2),

1

2|u − s|

)

one obtains the following formulae:

(3.16)

D(x,M) = Bn(y, r)

y =x(1 − t

2)

1 − |x|2t2 , r =(1 − |x|2)t1 − |x|2t2 , t = th 1

2M ,

and

(3.17)

Bn(x, a(1 − |x|)

)⊂ D(x,M) ⊂ B

n(x, A(1 − |x|)

),

a =t(1 + |x|)1 + |x|t , A =

t(1 + |x|)1 − |x|t , t = th 1

2M .

We shall often need a special case of (3.16):

(3.18) D(0,M) = Bn(th

1

2M) .


A standard application of formula (3.18) is the following observation. Let Tx bein M(Bn) as defined in 2.19 with Tx(x) = 0. Fix x, y ∈ Bn and z ∈ J [x, y] withρ(z, x) = ρ(z, y) = 1

2ρ(x, y). Then Tz(x) = −Tz(y) and (3.18) yields

(3.19)

|Tx(y)| = th 1

2ρ(x, y) ,

|Tz(x)| = th 1

4ρ(x, y) .

For an open set D in Rn, D 6= R

n, define d(z) = d(z, ∂D) for z ∈ D and

jD(x, y) = log(1 +

|x − y|mind(x), d(y)

)(3.20)

for x, y ∈ D. Then it is well-known that jD is a metric (see, e.g. [S]).

3.21. Lemma. The following inequalities

(1) jD(x, y) ≥∣∣∣ log

d(x)

d(y)

∣∣∣ ,

(2) jD(x, y) ≤∣∣∣ log

d(x)

d(y)

∣∣∣+ log(1 +

|x − y|d(x)

)≤ 2 jD(x, y)

(3) jD(x, y) ≥∣∣∣ log

|x − z||y − z|

∣∣∣

hold for all x, y ∈ D, z ∈ ∂D .

In the next lemma we show that jD yields simple two–sided estimates for ρD

both when D = Bn and when D = Hn.

3.22. Lemma. (1) jBn(x, y) ≤ ρBn(x, y) ≤ 4 jBn(x, y) for x, y ∈ Bn.

(2) jHn(x, y) ≤ ρHn(x, y) ≤ 2 jHn(x, y) for x, y ∈ Hn.

4. Quasihyperbolic geometry

In an arbitrary proper subdomain D of Rn one can define a metric, the quasi-

hyperbolic metric of D, which shares some properties of the hyperbolic metricof Bn or Hn. We shall now give the definition of the quasihyperbolic metric andstate without proof some of its basic properties which we require later on. Thequasihyperbolic metric has been systematically developed and applied by F. W.Gehring and his collaborators.

Throughout this section D will denote a proper subdomain of Rn. In D we

define a weight function w : D → R+ by

w(x) =1

d(x, ∂D); x ∈ D .(4.1)

Using this weight function one defines the quasihyperbolic length ℓq(γ) = ℓDq (γ)

of a rectifiable curve γ by a formula similar to (3.2). The quasihyperbolic distance

between x and y in D is defined by

kD(x, y) = infα∈Γxy

ℓDq (α) = inf

α∈Γxy

∫

α

w(x)|dx| ,(4.2)


where Γxy is as in (3.4). It is clear that kD is a metric on D. It follows from (4.2)that kD is invariant under translations, stretchings, and orthogonal mappings.(As in (3.3) one can define the quasihyperbolic volume of a (Lebesgue) measurableset A ⊂ D, but we shall not make use of this notion.) Given x, y ∈ D there existsa geodesic segment JD[x, y] of the metric kD joining x and y (cf. [GO]). However,very little is known about the structure of such geodesic segments JD[x, y] whenD is given. For some elementary domains, the geodesics were recently studiedby H. Linden [L].

4.3. Remarks. Clearly, kHn = ρHn , and we see easily that ρBn ≤ 2 kBn ≤ 2 ρBn

(cf. (4.1)). Hence, the geodesics of (Hn, kHn) are those of (Hn

, ρHn), but it is adifficult task to find the geodesics of kD when D is given. The following monotoneproperty of kD is clear: if D and D

′ are domains with D′ ⊂ D and x, y ∈ D

′,then kD′(x, y) ≥ kD(x, y).

In order to find some estimates for kD(x, y) we shall employ, as in the case ofHn and Bn, the metric jD defined in (3.20). The metric jD is indeed a naturalchoice for such a comparison function since both kD and jD are invariant undertranslations, stretchings and orthogonal mappings. A useful inequality is ([GP,Lemma 2.1])

kD(x, y) ≥ jD(x, y) ; x, y ∈ D .(4.4)

In combination with 3.22, (4.4) yields

kD(x, y) ≥∣∣∣ log

d(x)

d(y)

∣∣∣ , d(z) = d(z, ∂D) .(4.5)

For easy reference we record Bernoulli’s inequality

log(1 + as) ≤ a log(1 + s) ; a ≥ 1 , s > 0 .(4.6)

4.7. Lemma. (1) If x ∈ D, y ∈ Bx = Bn(x, d(x)), then

kD(x, y) ≤ log(1 +

|x − y|d(x) − |x − y|

).

(2) If s ∈ (0, 1) and |x − y| ≤ s d(x), then

kD(x, y) ≤ 1

1 − sjD(x, y) .

5. Modulus and capacity

For the definition and basic properties of the modulus we refer the reader to[V1]. The main sources for this section are [V1], [Vu3], [AVV2].

One of the main reasons why the modulus of a curve family is studied isthat we have a simple rule of transformation for the modulus of a curve familyunder quasiconformal mappings. Further, we would like to use modulus as aninstrument so as to gain insight about ”the geometry”. Roughly speaking wecan say that the modulus of the family of all curves joining two connected non-intersecting continua E,F ⊂ R

n behaves like mind(E), d(F )/d(E,F ) , where


d stands for the euclidean diameter. A long series of estimates is needed toreach this conclusion and its variants and some of these estimates are givenin this and the following section. Some of these variants involve hyperbolic orquasihyperbolic metric. In this fashion we step by step approach our goal, thestudy of how quasiconformal mappings between metric spaces deform distances.

5.1. Lemma. The p–modulus Mp is an outer measure in the space of all curve

families in Rn. That is,

(1) Mp(∅) = 0 ,

(2) Γ1 ⊂ Γ2 implies Mp(Γ1) ≤ Mp(Γ2) ,

(3) Mp

( ∞⋃

i=1

Γi

)≤

∞∑

i=1

Mp(Γi) .

Let Γ1 and Γ2 be curve families in Rn. We say that Γ2 is minorized by Γ1 and

write Γ2 > Γ1 if every γ ∈ Γ2 has a subcurve belonging to Γ1.

5.2. Lemma. Γ1 < Γ2 implies Mp(Γ1) ≥ Mp(Γ2).

5.3. Remark. The family of all paths joining E and F in G is denoted by∆(E,F ; G) see [Vu3, p.51]. If G = R

n or Rn

we often denote ∆(E,F ; G) by∆(E,F ). Curve families of this form are the most important for what follows.The following subadditivity property is useful. If E =

⋃∞j=1

Ej and cE(F ) =

Mp

(∆(E,F )

)= cF (E), then cF (E) ≤ ∑

cF (Ej), see 5.1(3). More precisely if

G ⊂ Rn

is a domain and F ⊂ G is fixed, then cGF (E) = Mp

(∆(E,F ; G)

)is an

outer measure defined for E ⊂ G. In a sense which will be made precise lateron, cE(F ) describes the mutual size and location of E and F . Assume now thatD is an open set in R

nand that F ⊂ D. It follows from 5.1(2) that

Mp

(∆(F, ∂D; D \ F )

)≤ Mp

(∆(F, ∂D; D)

)≤ Mp

(∆(F, ∂D)

).

On the other hand, because ∆(F, ∂D; D) < ∆(F, ∂D) and ∆(F, ∂D; D \ F ) <

∆(F, ∂D; D) , 5.2 yields

Mp

(∆(F, ∂D)

)= Mp

(∆(F, ∂D; D)

)= Mp

(∆(F, ∂D; D \ F )

).(5.4)

5.5. Lemma. Let D and D′ be domains in R

nand let f : D → D

′ be a conformal

mapping. Then M(fΓ) = M(Γ) for each curve family Γ in D where fΓ = f γ :γ ∈ Γ .5.6. Lemma. Let p > 1 and let E and F be subsets of R

n+. Then

Mp

(∆(E,F ; Rn

+))≥ 1

2Mp

(∆(E,F )

).

5.7. Corollary. Let E and F be sets in Rn

with q( E,F ) ≥ a > 0. Then

M(∆(E,F )

)≤ c(n, a) < ∞.

5.8. Lemma. (1) Let 0 < a < b and let E, F be sets in Rn with

E ∩ Sn−1(t) 6= ∅ 6= F ∩ S

n−1(t)


for t ∈ (a, b). Then

M(∆( E,F ; Bn(b) \ B

n(a) ))≥ cn log

b

a.

Equality holds if E = (ae1, be1), F = (−be1,−ae1). Here cn > 0 depends only on

n (see [V1, (10.11),(10.4)]).

(2) Let 0 < a < b . Then

M(∆( S

n−1(a), Sn−1(a); Bn(b) \ Bn(a) )

)= ωn−1(log(b/a))1−n

,

where ωn−1 is the (n − 1)-dimensional surface area of Sn−1

.

5.9. Corollary. If E and F are non–degenerate continua with 0 ∈ E ∩ F then

M(∆(E,F )

)= ∞.

Proof. Apply 5.8 with a fixed b such that Sn−1(b) ∩ E 6= ∅ 6= S

n−1(b) ∩ F andlet a → 0.

5.10. Canonical ring domains. A domain (open, connected set) D in Rn is

called a ring domain or, briefly, a ring, if Rn \ D consists of two components

C0 and C1. Sometimes we denote such a ring by R(C0, C1). In our study twocanonical ring domains will be of particular importance. These are the Grotzsch

ring RG,n(s), s > 1, and the Teichmuller ring RT,n(t), t > 0, defined by

(5.11)

RG,n(s) = R(Bn, [se1,∞]), s > 1,RT,n(t) = R([−e1, 0], [te1,∞]), t > 0.

Sometimes we also use the bounded Grotzsch ring R(Rn \ Bn, [0, re1]) . An im-

portant conformal invariant associated with a ring is the modulus of the familyof curves joining its complementary components. In the case of Grotzsch ringRG,n(s) and Teichmuller ring RT,n(t) the modulus is denoted by γn(s) and τn(t)respectively. It is a well-known basic fact that γn : (1,∞) → (0,∞) is a decreas-ing homeomorphism and that for all s > 1

(5.12) γn(s) = 2n−1τn(s2 − 1) .

Bn

s e 1 ∞ - e 1 0 t e1 ∞

R (s)G,ncap = M( ) = R (t)T,ncap = M( t

) = n(t)

s t

n(s)s

Figure 9. The Grotzsch and Teichmuller rings.


-1 1- K K

t- t 1 1-1

z w

-1 1- t t

z w

f-1

i K '/2

- r r

w = r sn( ,r), =2 i K

π Log zt

+ K- r r

Figure 10. Conformal map of an annulus onto a disk minus asymmetric slit.

t- t 1 1-1

zf

-1 - r r 1-1

wg

0 a

g f

Figure 11. Conformal map of an annulus onto a boundedGrotzsch ring.

5.13. Elliptic integrals and γ2(s). In the plane every ring domain can beconformally mapped onto an annulus z ∈ C : 1 < |z| < M for some M . Forthe Grotzsch ring this conformal mapping is given by the elliptic sn-function[AVV2]. For more information on the involved special functions see [QV].

As shown in [LV, II.2]

(5.14) γ2(s) = 2π/µ(1/s)

for s > 1 where

µ(r) =π

2

K(√

1 − r2 )

K(r), K(r) =

∫1

0

[(1 − x2)(1 − r

2x

2)]−1/2dx

for 0 < r < 1. The function K(r) is called a complete elliptic integral of the firstkind and its values can be found in tables.


0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

K K’

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

µ

Figure 12. The functions K(r) and µ(r) .

The modulus µ(r) satisfies the following three functional identities

(5.15)

µ(r)µ(√

1 − r2)

= 1

4π

2,

µ(r)µ(

1−r1+r

)= 1

2π

2,

µ(r) = 2µ(

2√

r

1+r

).

From (5.15) one can derive several estimates for µ(r) [LV, p. 62]. By [LV, p. 62]the following inequalities hold

(5.16) log1

r< log

1 + 3√

1 − r2

r< µ(r) < log

4

r

for 0 < r < 1. From (5.16) it follows that limr→0+ µ(r) = ∞ whence, by virtue ofthe functional identities (5.15), limr→1− µ(r) = 0. Therefore, µ : (0, 1) → (0,∞)is a decreasing homeomorphism. For the sake of completeness we set µ(0) = ∞and µ(1) = 0. By (5.14) and (5.15) we obtain

(5.17) γ2(s) =4

πµ

(s − 1

s + 1

), s > 1 .

5.18. Exercise. In the study of distortion theory of quasiconformal mappingsin Section 7 below the following special function will be useful

ϕK,n(r) =1

γ−1n (Kγn(1/r))


for 0 < r < 1, K > 0. (Note: [Vu1, Lemma 7.20] shows that γn is strictlydecreasing and hence that γ

−1

n exists.) Show that ϕAB,n(r) = ϕA,n(ϕB,n(r)) andϕ−1

A,n(r) = ϕ1/A,n(r) and that

ϕK,2(r) = ϕK(r) = µ−1(

1

Kµ(r)

).

Verify also that

(1) ϕ2(r) = 2√

r

1+r

(2) ϕK(r)2 + ϕ1/K

(√1 − r2

)2= 1 .

Exploiting (1) and (2) find ϕ1/2(r). Show also that

(3) ϕ1/K

(1−r1+r

)= 1−ϕK(r)

1+ϕK(r),

(4) ϕK

(2√

r

1+r

)=

2

√ϕK(r)

1+ϕK(r).

Lemma 7.22 in [Vu3] yields the inequalities

ωn−1 (log λns)1−n ≤ γn(s) ≤ ωn−1(log s)1−n

,(5.19)

ωn−1 (log(λ2

ns))1−n ≤ τn(s − 1) ≤ ωn−1(log s)1−n

,(5.20)

for s > 1.

5.21. Theorem. The function gn(t) = (ωn−1/γn(t))1/(n−1)−log t is an increasing

function on (1,∞) with limt→∞ gn(t) = log λn where λn ∈ [4, 2en−1), λ2 = 4 , is

so-called Grotzsch ring constant.

5.22. Theorem. For s ∈ (1,∞) and n ≥ 2

(1) γn(s) ≤ ωn−1µ(1/s)1−n< ωn−1

(log(s + 3

√s2 − 1 )

)1−n,

(2) 2n−1cn log

(s + 1

s − 1

)≤ γn(s) ≤ 2n−1

cn µ

(s − 1

s + 1

)< 2n−1

cn log(4

s + 1

s − 1

).

Moreover, if s ∈ (0,∞) and a = 1 + 2(1 +√

1 + s )/s, then

(3) cn log a ≤ τn(s) ≤ cnµ(1/a) < cn log(4a)

and (1 + 1/√

s )2 ≤ a ≤ (1 + 2/√

s )2 hold true. Furthermore, when n = 2, the

first inequality in (1), the second inequality in (2), and the second inequality in

(3) hold as identities.

5.23. Hyperbolic metric and capacity. As in Section 3 we let J [x, y] denotethe geodesic segment of the hyperbolic metric joining x to y, x, y ∈ Bn. It isclear by conformal invariance that

cap(Bn, J [x, y]) = cap(Bn

, TxJ [x, y])


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

f

g

F

G

Figure 13. Bounds for γ3 .

where Tx is as defined in 2.19. We obtain by (3.19) and [Vu1, (7.25)]

cap(Bn, J [x, y]) = γn

( 1

th 1

2ρ(x, y)

)≤ ωn−1

(− log th

1

4ρ(x, y)

)1−n.(5.24)

Next by (5.24), 5.22(2), and (5.16) we get

2n−1cn ρ(x, y) ≤ cap(Bn

, J [x, y]) ≤ 2n−1cn µ(e−ρ(x,y))

< 2n−1cn(ρ(x, y) + log 4) .(5.25)

For large values of ρ(x, y) (5.25) is quite accurate. For small ρ(x, y) one obtainsbetter inequalities than (5.25) by combining 5.22(1) and (5.24).

It is left as an easy exercise for the reader to derive from (5.19) the followinginequality

(5.26) tα/λn ≤ γ

−1

n (Kγn(t)) ≤ λαn t

α

for all t > 1 and K > 0, where α = K1/(1−n). From (5.26) it follows immediately

that

(5.27) rαλn

−α ≤ ϕK,n(r) ≤ λnrα

holds for all K > 0 and r ∈ (0, 1). For K ≥ 1 this inequality can be refined if weuse Theorem 5.22 (1).


5.28. Theorem. For n ≥ 2, K ≥ 1, and 0 ≤ r ≤ 1

(1) ϕK(r) ≤ λ1−αn r

α, α = K

1/(1−n),

(2) ϕ1/K(r) ≥ λ1−βn r

β, β = K

1/(n−1).

A compact set E ⊂ Bn is said to be of capacity zero, denoted capE = 0 , ifM(∆(E, S

n−1(2))) = 0 . A compact set E ⊂ Rn

is said to be of capacity zero, if itcan be mapped by a Mobius transformation onto a set E1 ⊂ Bn of capacity zero.Sets of capacity zero are very small: they have zero Hausdorff dimension, see[Vu3, p.86]. For many purposes they are negligible. The next theorem providesa convenient tool for estimating moduli of curve families in terms is geometricquantities and a set function.

5.29. Theorem. For n ≥ 2 there exist positive numbers d1, . . . , d4 and a set

function c(·) in Rn

such that

(1) c(E) = c(hE) whenever h : Rn → R

nis a spherical isometry and E ⊂ R

n.

(2) c(∅) = 0, A ⊂ B ⊂ Rn

implies c(A) ≤ c(B) and c

(⋃∞j=1

Ej

)

≤ d1

∑∞j=1

c(Ej) if Ej ⊂ Rn.

(3) If E ⊂ Rn

is compact, then c(E) > 0 if and only if cap E > 0. Moreover

c(Rn) ≤ d2 < ∞.

(4) c(E) ≥ d3 q(E) if E ⊂ Rn

is connected and E 6= ∅.(5) M

(∆(E,F )

)≥ d4 min c(E), c(F ) , if E,F ⊂ R

n.

Furthermore, for n ≥ 2 and t ∈ (0, 1) there exists a positive number d5 such that

(6) M(∆(E,F )

)≤ d5 minc(E), c(F ) whenever E,F ⊂ R

nand q(E,F ) ≥ t.

6. Conformal invariants

In the preceding sections we have studied some properties of the conformalinvariant M

(∆(E,F ; G)

). In this section we shall introduce two other conformal

invariants, the modulus metric µG(x, y) and its ”dual” quantity λG(x, y), whereG is a domain in R

nand x, y ∈ G. The modulus metric µG is functionally related

to the hyperbolic metric ρG if G = Bn, while in the general case µG reflects the“capacitary geometry” of G in a delicate fashion. The dual quantity λG(x, y)is also functionally related to ρG if G = Bn. As shown in [Vu3] for a wideclass of domains in R

n, the so–called QED–domains[GM], two–sided estimatesfor λG(x, y) in terms of

rG(x, y) =|x − y|

min d(x, ∂G), d(y, ∂G) .

6.1. The conformal invariants λG and µG. If G is a proper subdomain ofR

n, then for x, y ∈ G with x 6= y we define

λG(x, y) = infCx,Cy

M(∆(Cx, Cy; G)

)(6.2)

where Cz = γz[0, 1) and γz : [0, 1) → G is a curve such that γz(0) = z andγz(t) → ∂G when t → 1, z = x, y. It follows from 5.5 that λG is invariant under


conformal mappings of G. That is, λfG(f(x), f(y)) = λG(x, y), if f : G → fG isconformal and x, y ∈ G are distinct.

y

C x

C y

G

G(x,y)

y

G

G (x,y)

•

•

•

• xx

Cxy

Figure 14. The conformal invariants λG and µG .

6.3. Remark. If card(Rn \ G) = 1, then λG(x, y) ≡ ∞ by 5.9. Therefore λG is

of interest only in case card(Rn \ G) ≥ 2. For card(R

n \ G) ≥ 2 and x, y ∈ G,x 6= y, there are continua Cx and Cy as in (6.2) with Cx ∩ Cy = ∅ and thus

M(∆(Cx, Cy; G)

)< ∞ by 5.7. Thus, if card(R

n \ G) ≥ 2, we may assume that

the infimum in (6.2) is taken over continua Cx and Cy with Cx ∩ Cy = ∅.6.4. The extremal problems of Grotzsch and Teichmuller. The Grotzschand Teichmuller rings arise from extremal problems of the following type, whichwere first posed for the case of the plane: Among all ring domains which separatetwo given closed sets E1 and E2, E1 ∩ E2 = ∅, find one whose module has thegreatest value.

Let E1 be a continuum and E2 consist of two points not separated by E1. Bythe conformal invariance of the modulus one may then suppose that E1 = S

1 andE2 = 0, r, 0 < r < 1. Then the extremal problem is solved by the boundedGrotzsch ring R(R2 \ B

2, [0, r]). In other words, cap(B2

, E) ≥ γ2(1/r), whereE ⊂ B

2 is any continuum joining the points 0 and r ∈ R. For details we referthe reader to [LV, Ch. II].

The following function is the solution of the generalization of the Teichmullerproblem to R

n. For x ∈ Rn \ 0, e1, n ≥ 2, define

(6.5) p(x) = infE,F

M((E,F )),

where the infimum is taken over all pairs of continua E and F in Rn

with 0, e1 ∈E, x,∞ ∈ F . Teichmuller applied a symmetrization method to prove that forn = 2,

p(x) ≥ p((1 + |x − e1|)e1)

with equality for x = (1 + |x − e1|)e1 . For more details, see [HV] and [SoV].

For a proper subdomain G of Rn

and for all x, y ∈ G define

µG(x, y) = infCxy

M(∆(Cxy, ∂G; G)

)(6.6)


−1 0 1 2 3 4

−1

−0.5

0

0.5

1

1.5

2

2.5

3

∞

E

F

0 1

x

Figure 15. The extremal problem of Teichmuller.

where the infimum is taken over all continua Cxy such that Cxy = γ[0, 1] and γ

is a curve with γ(0) = x and γ(1) = y. It is clear that µG is also a conformalinvariant in the same sense as λG. It is left as an easy exercise for the readerto verify that µG is a metric if cap∂G > 0. [Hint: Apply 5.3 and 5.29.] Ifcap∂G > 0, we call µG the modulus metric or conformal metric of G.

6.7. Remark. Let D be a subdomain of G. It follows from 5.3 and (5.4)that µG(a, b) ≤ µD(a, b) for all a, b ∈ D and λG(a, b) ≥ λD(a, b) for all dis-tinct a, b ∈ D. In what follows we are interested only in the non–trivial casecard(R

n \G) ≥ 2. Moreover, by performing an auxiliary Mobius transformation,we may and shall assume that ∞ ∈ R

n \ G throughout this section. Hence G

will have at least one finite boundary point.

In a general domain G, the values of λG(x, y) and µG(x, y) cannot be expressedin terms of well–known simple functions. For G = Bn they can be given in termsof ρ(x, y) and the capacity of the Teichmuller condenser.

6.8. Theorem. The following identities hold for all distinct x, y ∈ Bn:

(1) µBn(x, y) = 2n−1τ

(1

sh2 1

2ρ(x, y)

)= γ

(1

th 1

2ρ(x, y)

),

(2) λBn(x, y) =1

2τ

(sh2

1

2ρ(x, y)

).


6.9. Remark. (1) In [Vu3, p. 193] it was stated as an open problem, whetherλD(x, y)1/(1−n) is a metric when D = R

n \ 0 and n = 2 . Subsequently theproblem was solved by A. Solynin [So] and J. Jenkins [J] for n = 2 . J. Ferrand[F] proved that λD(x, y)1/(1−n) is a metric for all D ⊂ R

n, n ≥ 2.

(2) From 5.22(3) we obtain the following inequality for x, y ∈ Bn (exercise)

1

2τ

(sh2

1

2ρ(x, y)

)≥ −cn log th

1

4ρ(x, y)

= 2cn arth(e− 1

2ρ(x,y)

)≥ 2cne

− 1

2ρ(x,y)

.

Here the identities 2 ch2A = 1 + ch 2A , sh 2A = 2 ch A sh A , and log th s =

−2 arth e−2s were applied. Recall that

sh21

2ρ(x, y) =

|x − y|2(1 − |x|2)(1 − |y|2)

by (3.12). Similarly, by 5.22(3) we obtain also

1

2τ

(sh2

1

2ρ(x, y)

)≤ 1

2cn µ

(th2(

1

4ρ(x, y))

)<

1

2cn log

4

th2 1

4ρ(x, y)

= cn log2

th 1

4ρ(x, y)

.

6.10. Lemma. Let G be a proper subdomain of Rn, x ∈ G, d(x) = d(x, ∂G),

Bx = Bn(x, d(x)), let y ∈ Bx with y 6= x, and let r = |x − y|/d(x). Then the

following two inequalities hold:

(1) λG(x, y) ≥ λBx(x, y) =

1

2τ

(r2

1 − r2

)> cn log

1

r

(2) µG(x, y) ≤ µBx(x, y) = γ

(1

r

)≤ ωn−1

(log

1

r

)1−n

.

6.11. Lemma. The inequality

p(x) ≥ max

τ(|x|) , τ(|x − e1|)

holds for all x ∈ Rn \ 0, e1. Equality holds if x = se1 and s < 0 or s > 1.

The following theorem summarizes some properties of p(x).

6.12. Theorem. For |x − e1| ≤ |x|, x ∈ Rn \ 0, e1

(1) p(x) ≤ 2 τ(|x − e1|) when |x + e1| ≥ 2,

(2) p(x) ≤ 4 τ(|x − e1|) when |x| ≥ 1,

(3) p(x) ≤ 2n+1τ(|x − e1|).

This result was improved by D. Betsakos [B] who proved the next theorem.The sharp constant in Theorem 6.13 is not known for n > 2 , for n = 2, see [BV].

6.13. Theorem. For x ∈ Rn \ 0, e1

(6.14) p(x) ≤ 4τ(min|x|, |x − e1|) .


For x ∈ Rn\0 we denote by rx a similarity map with rx(0) = 0 and rx(x) = e1.

Then it is easy to see that |rx(y)− e1| = |x− y|/|x|. It follows immediately fromthe definitions (6.2) and (6.5) that

λRn\0(x, y) = min p(rx(y)) , p(ry(x)) .(6.15)

Next we deduce the following two–sided inequality for λRn\0(x, y).

6.16. Theorem. For distinct x, y ∈ Rn \ 0 the following inequality holds

1 ≤ λRn\0(x, y)/τ

(|x − y|/ min|x|, |y|

)≤ 4 .

6.17. Corollary. Let G be a proper subdomain of Rn, x and y distinct points

in G and m(x, y) = mind(x), d(y). Then

λG(x, y) ≤ infz∈∂G

λRn\z(x, y) ≤ 4 τ

(|x − y|/m(x, y)

).

Proof. The first inequality follows from 6.7. For the second one fix z0 ∈ ∂G withm(x, y) = d(x, y, z0). Applying 6.16 to R

n \ z0 yields the desired result.

We next show that 6.17 fails to be sharp for a Jordan domain G in Rn. For

t ∈ (0, 1

5) consider the family Gt = B

n(−e1, 1) ∪ Bn(e1, 1) ∪ B

n(t) of Jordandomains. Then by 6.17

λGt(−e1, e1) ≤ 4 τ(2)

for all t ∈ (0, 1

5). But this is far from sharp because in fact

λGt(−e1, e1) ≤ M

(∆( [−2e1,−e1] , [e1, 2e1] ; Gt)

)

≤ ωn−1

(log

1

t

)1−n

−→ 0

as t → 0. However, for a wide class of domains, which we shall now consider,the upper bound in 6.17 is essentially best possible.

−e1 e1

Figure 16. The family ∆( [−2e1,−e1] , [e1, 2e1] ; Gt) .


6.18. QED domains. A closed set E in Rn

is called a c–quasiextremal distance

set or c-QED exceptional set or c-QED set, c ∈ (0, 1], if for each pair of disjointcontinua F1, F2 ⊂ R

n \ E

M(∆(F1, F2; R

n \ E))≥ c M

(∆(F1, F2)

).(6.19)

If G is a domain in Rn

such that Rn \G is a c-QED set, then we call G a c-QED

domain.

6.20. Examples. (1) The unit ball Bn is a 1

2–QED set by [GM1] or by the

above Lemma 5.6.

(2) If E is a compact set of capacity zero, then E is a 1–QED set. For instanceall isolated sets are 1–QED sets. The class of all 1–QED sets contains all closedsets in R

n of vanishing (n−1)–dimensional Hausdorff measure (see [V3], [GM1]).

(3) B2 \ [0, e1) is not a c-QED set for any c > 0.

6.21. Theorem. Let G be a c-QED domain in Rn. Then

λG(x, y) ≥ c τ(s2 + 2s) ≥ 21−nc τ(s)

where s = |x − y|/ mind(x), d(y).

It should be noted that the lower bound of 6.21 is very close to that of 6.16;in fact it differs only by a multiplicative constant.

In the next few theorems we shall give some estimates for the conformalmetric µG.

6.22. Lemma. Let G be a proper subdomain of Rn, s ∈ (0, 1), x, y ∈ G. If

kG(x, y) ≤ 2 log(1 + s), then

(1) µG(x, y) ≤ γ

( 1

th(kG(x, y)/(1 − s))

).

Moreover, there exist positive numbers b1 and b2 depending only on n such that

(2) µG(x, y) ≤ b1kG(x, y) + b2

for all x, y ∈ G.

It should be observed that (6.22(2)) is a generalization of the upper bound in(5.25) to the case of an arbitrary domain. The lower bound in (5.25) will nextbe generalized to the case of domains with connected boundary.

6.23. Lemma. Let G be a domain in Rn such that ∂G is connected. Then for

all a, b ∈ G, a 6= b,

(1) µG(a, b) ≥ τ(4m2 + 4m) ≥ cn jG(a, b)

where cn is the constant in 5.8 and m = mind(a), d(b)/|a− b|. If, in addition,

G is uniform, then

(2) µG(a, b) ≥ B kG(a, b)

for all a, b ∈ G.


7. Distortion theory

For the basic properties and definitions of K-quasiconformal and K-quasiregularmappings we refer the reader to [Vu3] as well as to the other papers in this samevolume. See, in particular, Rasila’s paper. One of the key ideas is that undera K-quasiconformal mapping, the modulus is changed at most by a constantc ∈ [1/K,K] . The notions introduced in the previous chapters enable us to for-mulate this basic property in a more concrete and geometric way, in terms ofmetrics.

Theorem 7.1 and Corollary 7.2 are the key results of this paper, and the otherresults in this section are just consequences. One should carefully observe thatalthough the transformation rule in Corollary 7.2 looks like a bilipschitz property;the mappings need not be bilipschitz in the euclidean metric. This is because themetric µG behaves in a non-linear fashion. In the euclidean metric quasiconformalmappings are Holder-continuous as the results below show.

7.1. Theorem. If f : G → Rn is a non–constant qr mapping, then

(1) µfG(f(a), f(b)) ≤ KI(f) µG(a, b) ; a, b ∈ G .

In particular, f : (G,µG) → (fG, µfG) is Lipschitz continuous. If N(f,G) < ∞,

then

(2) λG(a, b) ≤ KO(f) N(f,G) λfG(f(a), f(b))

for all a, b ∈ G with f(a) 6= f(b).

7.2. Corollary. If f : G → G′ = fG is a qc mapping, then

(1) µG(a, b)/KO(f) ≤ µfG(f(a), f(b)) ≤ KI(f) µG(a, b) ,

(2) λG(a, b)/KO(f) ≤ λfG(f(a), f(b)) ≤ KI(f) λG(a, b)

hold for all distinct a, b ∈ G.

7.3. Theorem. Let f : Bn → Rn be a non–constant K–qr mapping with fBn ⊂

Bn and let α = KI(f)1/(1−n). Then

(1) th 1

2ρ

(f(x), f(y)

)≤ ϕK

(th 1

2ρ(x, y)

)≤ λ

1−αn

(th 1

2ρ(x, y)

)α,

(2) ρ

(f(x), f(y)

)≤ KI(f)

(ρ(x, y) + log 4

),

hold for all x, y ∈ Bn, where λn is the Grotzsch ring constant.

7.4. Corollary. Let f : Bn → Bn be a K–qr mapping with f(0) = 0 and let

α = KI(f)1/(1−n). Then

(1) |f(x)| ≤ ϕK,n(|x|) ≤ λ1−αn |x|α ≤ 21−1/K

K|x|1/K,

(2) |f(x)| ≤ a − 1

a + 1, a =

(4

1 + |x|1 − |x|

)KI(f)

,

for all x ∈ Bn.


7.5. Example. Let g : B2 → B2 \ 0 = gB2 be the exponential functiong(z) = exp( z+1

z−1), z ∈ B2. We shall show that g : (B2

, ρ) → (gB2, kgB2) fails to

be uniformly continuous. To this end, let xj = (ej − 1)/(ej + 1), j = 1, 2, . . . .Then it follows that ρ(0, xj) = j and thus ρ(xj, xj+1) = 1. Let Y = B2 \ 0.Since g(xj) = exp(−e

j) we get by (4.4) and (3.20)

kY

(g(xj), g(xj+1)

)≥ jY

(g(xj), g(xj+1)

)

= log[1 +

(exp e

j+1) (

exp(−ej) − exp(−e

j+1))]

= log[1 + exp(ej+1 − e

j) − 1]

= ej+1 − e

j → ∞as j → ∞. In conclusion, g : (B2

, ρ) → (Y, kY ) cannot be uniformly continuous,because ρ(xj, xj+1) = 1.

7.6. Theorem. Let f : Bn → Rn be a non–constant qr mapping, let E ⊂

Rn \ fBn be a non–degenerate continuum such that ∞ ∈ E, and let G = R

n \E

be a domain.

(1) Then f : (Bn, ρ) → (G, jG) is uniformly continuous.

(2) If G is uniform, then f : (Bn, ρ) → (G, kG) is uniformly continuous.

7.7. Theorem. Suppose that f : G → Rn is a bounded qr mapping and that F

is a compact subset of G. Let α = KI(f)1/(1−n) and C = λ1−αn d(fG)/d(F, ∂G)α

where λn is the Grotzsch constant. Then f satisfies the Holder condition

|f(x) − f(y)| ≤ C |x − y|α(7.8)

for x ∈ F , y ∈ G.

7.9. Theorem. Let f : Bn → Rn be a non–constant qr mapping.

(1) If ϕ ∈ (0, 1

2π) and fBn ⊂ C(ϕ), then for all x ∈ Bn

|f(x)| ≤ |f(0)| 4aϕ(1 + |x|

1 − |x|)aϕ

where a depends only on n and KI(f).(2) If fBn ⊂ x ∈ R

n : x2

1+ · · · + x

2

n−1< 1 , then for all x, y ∈ Bn

|f(x)| ≤ |f(y)| + AKI(f) (ρ(x, y) + log 4)

where A is a positive constant depending only on n.

7.10. Theorem. Let f : Bn → Bn be a qr mapping with N(f,Bn) = N < ∞.

Then

th 1

4ρ

(f(x), f(y)

)≤ 2

(th 1

4ρ(x, y)

)β

holds for all x, y ∈ Bn where β = 1/(NKO(f)). Furthermore, if f(0) = 0, then

for all x ∈ Bn

|f(x)|1 +

√1 − |f(x)|2

≤ 2( |x|

1 +√

1 − |x|2)β

.


7.11. Exercise. Assume that f : Bn → Bn is K–qc with f(0) = 0 and fBn =Bn. Show that

|f(x)|2 ≤ min

ϕ2

K,n(|x|) , 1 − ϕ2

1/K,n

(√1 − |x|2

) ,

|f(x)|2 ≥ max

ϕ2

1/K,n(|x|) , 1 − ϕ2

K,n

(√1 − |x|2

) .

Note that in the case n = 2 we have ϕ2

K,2(r) = 1−ϕ2

1/K,2(√

1 − r2 ) for all K > 0and 0 < r < 1, while the analogous relation fails to hold for n ≥ 3.

The next result is a famous theorem of A. Mori from 1956 [LV]. The theoremhas, however, one esthetic flaw: it is not sharp when K = 1 . It was conjecturedin the 1960’s that the constant 16 in the theorem could be replaced by 161−1/K

and also shown in [LV] that this would be sharp. In 1988 it was proven in [FeV]that we can replace 16 by M(K) → 1 as K → 1 . Perhaps the latest paperdealing with the problem of reducing the constant M(K) was written by S.-L.Qiu [Q], but as far as we know it is still an open problem whether the constant161−1/K could be achieved. Settling this problem would be remarkable progress,since a lot of work has been done. For the spherical chordal metric this problemwas recently discussed by P. Hasto [H3].

7.12. Theorem. Let f : B2 → B2 be a K–qc mapping with f(0) = 0 and

fB2 = B2. Then

|f(x) − f(y)| ≤ 16 |x − y|1/K

for all x, y ∈ B2. Furthermore, the number 16 cannot be replaced by any smaller

absolute constant.

7.13. An open problem. For K ≥ 1, n ≥ 2, and r ∈ (0, 1) let

ϕ∗K,n(r) = ϕ

∗K(r) = sup |f(x)| : f ∈ QCK(Bn), f(0) = 0, |x| ≤ r

where QCK(Bn) = f : Bn → fBn | f is K–qc and fBn ⊂ Bn . As shown in[LV, p. 64]

ϕ∗K,2(r) = ϕK,2(r) ≤ 41−1/K

r1/K(7.14)

for each r ∈ (0, 1) and K ≥ 1. By 7.4(1)

ϕ∗K,n(r) ≤ ϕK,n(r) ≤ λ

1−αn r

α, α = K

1/(1−n),(7.15)

for n ≥ 2, K ≥ 1, r ∈ (0, 1). A. V. Sychev [SY, p. 89] has conjectured that

ϕ∗K,n(r) ≤ 41−α

rα(7.16)

for all n ≥ 2 and K ≥ 1. Because λ2 = 4, (7.16) agrees with (7.14) for n = 2.In [AVV4] it is shown that ϕ

∗K,n 6≡ ϕK,n for n ≥ 3. It follows from 7.10 and 7.11

that

(7.17)

ϕ∗K,n(r) ≤ 4 r

1/K,[

ϕ∗K,n(r)

]2 ≤ 1 − ϕ2

1/K,n(√

1 − r2 ) .


From (7.17) and (7.15) it follows, as shown in [AVV], that

(7.18) ϕ∗K,n(r) ≤ 41−1/K2

r1/K

holds for all n ≥ 2, K ≥ 1, r ∈ (0, 1). Note that the right hand side of (7.18)is bounded when K → ∞. Recall that λn → ∞ as n → ∞ and that λ

1−αn ≤

21−1/KK. Note that Sychev’s conjecture (7.16) still remains open.

7.19. Another open problem. In [Vu4], the following problem was stated.Let QCK(Rn) = f : R

n → Rn | f is K–qc and f(e1) = e1 . Is it true that

(7.20) sup|g(x)| : |x| = r, g ∈ QCK(Rn) = sup|f(−re1)| : f ∈ QCK(Rn) ,

when r > 0? For n = 2 the answer is in the affirmative by [LVV].

8. Open problems

Assume that G ⊂ Rn is a proper subdomain. For what follows, we will be

interested mainly in the cases when the domain is a member of some well-knownclass of domains. Some examples are uniform domains, QED-domains, domainswith uniformly perfect (in the sense of Pommerenke [Su]) boundaries and qua-siballs, i.e. domains G of the form G = fBn for quasiconformal f : R

n → Rn.

We denote the class of domains with D . Let us consider collection of metricsM = αG, hG, jG, kG, λ

1/(1−n)

G , µG, q, | · | where hG refers to the hyperbolic met-ric when n = 2. Interesting categories of mappings, we denote them by C, areHolder, Lipschitz, isometries, quasiisometries and identity mappings.

The problems that we list below are just examples. There are a great manyvariations, by letting the domain, mapping and metric independently vary overthe categories D , C , and M .

8.1. Convexity of balls and smoothness of spheres. Fix m ∈ M . Doesthere exist constant T0 > 0 such that Dm(x, T ) = z ∈ G : m(x, z) < T, isconvex (in euclidean geometry) for all T ∈ (0, T0)? Is ∂Gm(x, T ) smooth forT < T0?

For instance, in the case m = kG both of these problems seem to be open. Inpassing, we remark that it follows from (4.4) and Theorem 4.7 (2) that whenthe radius tends to 0, quasihyperbolic balls become more and more round. Thequasihyperbolic metric is used as a tool for many applications, but very littleabout the metric itself is known. See the theses [MA] and [L] and also Linden’spaper in this volume.

8.2. Lipschitz-constant of identity mapping. For x, y ∈ Bn, x 6= y , the

following inequality holds [Vu3, (2.27)]

|x − y| ≤ 2 thρBn(x, y)

4<

ρBn(x, y)

2.

We may now regard this result as an inequality for the modulus of continuity ofid : (Bn

, ρBn) → (Bn, | · |) . Instead of considering the identity mapping we could

now take any mapping in our category of mappings and consider the problem


of estimating the modulus of continuity between any two metric spaces in ourcategory of metric spaces, see [Vu1], [S]. We list several particular cases of ourproblem. For G = R

n \ 0 does there exist constants A or B such that for allx, y ∈ G

q(x, y) ≤ AkG(x, y),

andq(x, y) ≤ Bλ

1/(1−n)

G (x, y) ?

For G = C \ 0, 1 does there exist constant C such that for all x, y ∈ G

q(x, y) ≤ ChG(x, y) ,

For G = Rn \ 0 does there exist a constant E such that for all x, y ∈ G

λ1/(1−n)

G (x, y) ≤ EjG(x, y) ?

8.3. Characterization of isometries and quasiisometries. Given two met-ric spaces in our category of spaces, does there exist a quasiisometry, mappingthe one space onto the other space? Again, we could consider, in place of quasi-isometries, any other map in our category of maps.

What is the modulus of continuity of id : (G,µG) → (G, λ1/(1−n)

G )?

Is a quasiisometry f : (G, λ1/(1−n)

G ) → (fG, λ1/(1−n)

fG ) quasiconformal? J. Lelong-Ferrand raised this question in [LF] and the question was answered in the nega-tive in [FMV] . There it was also shown that the answer is affirmative under the

stronger requirement that f : (D,λ1/(1−n)

D ) → (fD, λ1/(1−n)

fD ) be uniformly contin-uous for all subdomains D of G . However, it is not known what the isometriesare.

Are isometries f : (G,αG) → (fG, αfG) Mobius transformations? (see Beardon[BE2], Hasto and Ibragimov [HI] and also Hasto’s paper in this volume).

8.4. Conformal invariants. The conformal invariant p(x) is relatively well-known. See [HV] for further information. However, much less is known aboutthe invariants µG and λG . For domains whose boundaries are uniformly perfect(in the sense of Pommerenke), there are some inequalities for µG in terms of jG ,

see [Vu2] and [JV]. Some results for λG when G = Bn \ 0 , were proved in [H]and [BV]. But even the basic question of finding a formula for λB2\0(x, y) isopen.

Some of these problems may be hard, some are very easy. Because of thevery general setup, it would require some effort even to single out the interestingcombinations of domains in D , mappings in C , and metrics in M .

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Matti Vuorinen E-mail: [email protected]: University of Turku


Circle Packings, Quasiconformal Mappings, andApplications

G. Brock Williams

Abstract. We provide an overview of the connections between circle packingsand quasiconformal mappings, with particular attention to applications tostring theory and image recognition.

Keywords. Circle Packing, Quasiconformal Maps.

2000 MSC. 52C26, 30F60.

Contents

1. Introduction 328

2. Quasiconformal Maps 328

2.1. Analytic Definition of Quasiconformality 328

2.2. Geometric Definition of Quasiconformality 329

2.3. An Important Example 330

3. Conformal Welding 331

3.1. Quasisymmetries and Quasicircles 331

3.2. Conformal Welding Theorem 332

4. Circle Packing 333

4.1. Definitions and Examples 333

4.2. Packings and Maps 335

4.3. The Rodin-Sullivan Theorem 335

5. Applications 338

5.1. Image Recognition 338

5.2. Radnell-Schippers Quantum Field Theory 341

5.3. Circle Packing Measurable Riemann Mapping Theorem 342

References 343

Version June 10, 2006.Supported by NSF Grant #0536665.

328 G. Brock Williams IWQCMA05

1. Introduction

The deep connections between the combinatorial and geometric properties ofcircle packings and the analytic properties of the maps they induce have been thesubject of intense study in recent years. In 1985, William Thurston conjectured,and Burt Rodin and Dennis Sullivan proved, that maps between circle packingswere nearly analytic [Thu85,RS87]. Since then the study of circle packings hasexploded to impact a great many other fields including conformal mapping [HS93,HS96,Ste05], complex analysis [BS91,DS95a,Ste97,Ste02,Ste03] Teichmuller the-ory [BS90, Bro96, Wil01b, BW02, Wil03, BS04b], brain mapping [Bea99, Kra99],random walks [Ste96,Dub97,HS95,McC98,DW05], tilings [BS97,Rep98], mini-mal surfaces and integrable systems [BS04a], numerical analysis [Moh93,CS99],metric measure spaces [BK02] and much more.

The fundamental folk theorem of circle packing is that “packings desperatelywant to be conformal.” They react to combinatorial or geometric changes inprecisely the same way as conformal maps. Maps between packings seem de-termined to approximate conformal maps. There is, however, much to be saidabout the relationships between circle packings and quasiconformal maps. It isprincipally with these connections and the applications arising from them thatwe will concern ourselves in this paper.

After some initial background on quasiconformal mappings in Section 2, wedescribe the crucial concept of conformal welding in Section 3. We review thefundamental concepts of circle packing in Section 4, and then describe threeapplications of circle packings and quasiconformal maps in Section 5. Namely,we discuss the use of packings in image recognition, in implementing Radnell-Schippers quantum field theory, and in constructing quasiconformal maps.

2. Quasiconformal Maps

2.1. Analytic Definition of Quasiconformality. Quasiconformal mappingsform the heart of Teichmuller theory as developed in the 1950’s and 1960’s.They are the natural generalization of analytic functions. For more detailedexplanations, a number of excellent resources are available, including [Ahl66,LV73,Leh87,Nag88, IT92,GL00].

Definition 2.1. A homeomorphism f ∈ L2 is quasiconformal if

(2.1) ∂zf = µ∂zf

for some µ ∈ L∞, ||µ||∞ < 1. Recall the complex partial derivatives are defined

by

∂zf =1

2(∂xf + ∂yf)

∂zf =1

2(∂xf − ∂yf) .

Circle Packings, QC Maps, and Applications 329

f

Figure 1. A geometric measure of quasiconformality. The quo-tient of the length of the dashed lines measures how close the curveon the right is to being a circle.

Equation 2.1 is called the Beltrami equation and µ, a Beltrami differ-ential. Notice that when µ ≡ 0, the Beltrami equation becomes ∂zf ≡ 0,which when separated into real and imaginary parts is precisely the familiarCauchy-Riemann equations. Thus µ determines how “quasi” a quasiconformalmap really is. This measure of the “quasi-ness,” or distortion of a map is mostoften expressed in terms of the dilatation

K =1 + ||µ||∞1 − ||µ||∞

≥ 1

of the map. A quasiconformal map f with dilatation K is called a K-quasiconformalmap; a 1-quasiconformal map is thus conformal.

The Beltrami differential µ corresponding to a quasiconformal map is oftencalled its complex dilatation. Notice, however, that the complex dilatation isa complex function and actually measures the distortion of f at every point in itsdomain. The dilatation, on the other hand, is a single real number and providesa global bound on the distortion of f over the entire domain.

2.2. Geometric Definition of Quasiconformality. An equivalent measureof the distortion of a quasiconformal map is provided by the dilatation quotient

Df (z) = lim supr→0+

supθ|f(z + reiθ) − f(z)|

infθ|f(z + reiθ) − f(z)| .

The dilatation quotient has a simple geometric interpretation. If we consider asmall circle of radius r about z in the domain, it will be mapped to some curveabout f(z) in the range. The dilatation quotient is then the ratio of the maximalto the minimal distance from f(z) to this curve. See Figure 1.

Recall that conformal maps preserve angles; moreover, if f′(z) = re

iθ 6= 0,then

df = f′(z) dz = re

iθdz.

Thus infinitesimally, f acts geometrically like

z 7→ reiθz + C


for some number C; that is, f acts like the composition of a scaling, rotation, andtranslation. This not only explains the reason analytic maps with non-vanishingderivative preserve angles, but also implies that they must map infinitesimal cir-cles to infinitesimal circles. Consequently, the dilatation quotient of a conformalmap is identically 1.

It turns out that the dilatation of a quasiconformal map is nothing more thanthe supremum of the dilatation quotient over the domain. Thus we have thefollowing equivalent definition of quasiconformality.

Definition 2.2. A homeomorphism f is K-quasiconformal if it is absolutelycontinuous on lines and

Df (z) ≤ K

for all z in its domain.

2.3. An Important Example. If we think of the complex plane as R2 and

x + iy as

(x

y

), then it is natural to consider the effect of linear and affine

transformations. Suppose

(2.2) f(x + iy) =

(a b

c d

) (x

y

)+

(e

f

),

where ad − bc 6= 0.

A moment’s linear algebra shows f can be re-written as

(2.3) f(x + iy) =

(ax + by + e

cx + dy + f

).

Then

∂zf =1

2

((a

c

)−

(−d

b

))=

1

2

(a + d

c − b

)(2.4)

∂zf =1

2

((a

c

)+

(−d

b

))=

1

2

(a − d

c + b

).

Notice that ∂zf = 0 if and only if a = d and c = −b, in which case, mul-

tiplication by the matrix

(a b

c d

)is equivalent to multiplication by the complex

number a + ib.

In general, however, we will have

µ =∂zf

∂zf=

(a − d) + i(c + b)

(a + d) + i(c − b),

and f will be quasiconformal.

Geometrically, f will map the basis vectors 1 and i to a + ic and b + id,respectively, and then translate by e + if . It is easy to check that µ = 0 if and


only if the new basis vectors a+ ic and b+ id are perpendicular, and |µ| increasestoward 1 as the angle decreases toward 0.

Notice that affine maps have constant complex dilatation; conversely, if µ isconstant, it is a simple exercise to solve for the affine map whose dilatation is µ.

The importance of this example becomes apparent when we consider the infin-itesimal behavior of any quasiconformal map. Just as we observed that confor-mal maps act infinitesimally by rotation, scaling, and translation, quasiconformalmaps act infinitesimally as affine maps.

3. Conformal Welding

3.1. Quasisymmetries and Quasicircles. We continue our exploration ofquasiconformal maps with an investigation of their boundary values [BA56,LV73,DE86, LP88,GL00]. Note that when maps extend continuously or smoothly tothe boundary, we will use same notation for the extended maps.

Definition 3.1. A homeomorphism ϕ : ∂D → ∂D is quasisymmetric or aquasisymmetry if it is the boundary function of some quasiconformal map ofD onto itself.

As might be expected, quasisymmetries have a beautiful geometric character-ization as well [BA56,LV73,Leh87,Krz87].

Definition 3.2. An orientation preserving homeomorphism ϕ : ∂D → ∂D is ak - quasisymmetry if

1

k≤ |ϕ(I)|D

|ϕ(J)|D≤ k

for any two adjacent intervals (subarcs) I and J of ∂D having equal length|I|D = |J |D.

Essentially, this definition says quasisymmetries can’t map adjacent symmetricintervals to extremely non-symmetric intervals.

Next, we temporarily leave quasisymmetries to consider the effect of quasicon-formal maps on circles. However, as we will see, these quasicircles are intimatelyconnected to quasisymmetries.

Definition 3.3. A Jordan curve Γ is a K-quasicircle if it is the image of theunit circle under a K-quasiconformal map of C onto itself.

As might be expected by now, quasicircles have both analytic and geometricdefinitions [Ahl63,Ahl66].

Definition 3.4. A Jordan curve Γ is a quasicircle if there exists R > 1 so thatfor all points x, y ∈ Γ

diam(Γx,y) ≤ R|x − y|,where Γx,y is the sub-arc of Γ connecting x and y which has the smaller diameter.


g

f

Figure 2. If Γ is a Jordan curve, then the Riemann MappingTheorem promises the existence of a conformal map f from theinside of Γ to the inside of the unit disc D. Similarly, there existsa conformal map g from the outside of Γ to the outside of the unitdisc.

g

f −1

Figure 3. Since f and g extend to the boundary, they induce ahomeomorphism ϕ = g f

−1 : ∂D → ∂D.

Loosely speaking, this condition limits “pinching” – a quasicircle cannot visit apoint x, wander far away, and then return to a point very near x. Fred Gehring’smonograph [Geh82] contains an extensive list of these and other characterizationsof quasicircles.

3.2. Conformal Welding Theorem. The intimate connection between qua-sisymmetries and quasicircles is illustrated by the following two theorems [Pfl51,LV73,Leh87,GL00].

Theorem 3.5. Suppose Γ is Jordan curve dividing the plane into complemen-

tary components Ω and Ω∗. Let f : Ω → D and g : Ω∗ → D∗ be conformal

homeomorphisms, the existence of which are promised by the Riemann Mapping

Theorem. Then f and g extend to homeomorphisms of the boundary and

g f−1 : ∂D → ∂D

is a quasisymmetry if Γ is a quasicircle. See Figures 2 and 3.


The converse is also true. Given a quasisymmetry ϕ : ∂D → ∂D, we can glue D

and D∗ together by attaching points e

iθ ∈ ∂D to their image points ϕ

(e

iθ)∈ ∂D

∗.The result is a topological sphere. As D and D

∗ struggle to fit together after thewelding, the “seam” between them will be pushed and pulled into a quasicircle.

Conformal Welding Theorem. Let ϕ : ∂D → ∂D be a quasisymmetry. Then

ϕ induces a conformal welding of D and D∗. That is, there exist conformal

maps f : Ω → D and g : Ω∗ → D∗ of complementary Jordan domains in C with

boundary values satisfying

g f−1(eiθ) = ϕ(eiθ).

Moreover, the Jordan curve Γ = f−1(∂D) = g

−1(∂D∗) is unique up to Mobius

transformations.

For quasisymmetries defined on ∂D, it is customary to normalize our weldingmaps so that f

−1(1) = g−1(1) = ϕ(1) = 1, f(0) = 0, and g(∞) = ∞. With these

normalizations, the maps f and g and the curve Γ are unique.

4. Circle Packing

4.1. Definitions and Examples. Since William Thurston’s work in the mid-1980’s, the connections between circle packings and analytic functions have beenwidely studied. More detailed information is contained in the rapidly expandingliterature, including several recent survey articles [DS95b,Ste97,Ste02,Ste03] andKen Stephenson’s excellent new book [Ste05].

Definition 4.1. A CP-complex K is an abstract simplicial 2-complex suchthat

1. K is simplicially equivalent to a triangulation of an (orientable) surface.2. Every boundary vertex of K has an interior neighbor.3. The collection of interior vertices is nonempty and edge-connected.4. There is an upper bound on the degree of vertices in K.

The restrictions imposed by conditions 2 through 4 are extremely mild andare met by most any reasonable triangulation.

Notice that a CP-complex is a purely combinatorial object. It possesses nogeometric structure until it is embedded in a surface by a circle packing. Toemphasize this fact, we will often refer to a CP-complex simply as an abstracttriangulation.

Definition 4.2. A circle packing is a configuration of circles with a specifiedpattern of tangencies. In particular, if K is a CP-complex, then a circle packingP for K is a configuration of circles such that

1. P contains a circle Cv for each vertex v in K,2. Cv is externally tangent to Cu if [v, u] is an edge of K,3. 〈Cv, Cu, Cw〉 forms a positively oriented mutually tangent triple of circles if

〈v, u, w〉 is a positively oriented face of K.


Figure 4. A finite circle packing (left). The underlying trian-gulation can be recovered by connecting centers of tangent circleswith line segments (middle). The resulting collection of trianglesforms the carrier of the packing (right).

Figure 5. A portion of the “regular hex” packing. Notice thatevery circle has the same radius.

A packing is called univalent if none of its circles overlap, that is, if no pair ofcircles intersect in more than one point.

A univalent circle packing produces a geometric realization of its underlyingcomplex. Vertices can be embedded as centers of their corresponding circles,and edges can be realized as geodesic segments joining centers of circles. Thecollection of triangles embedded in this way is called the carrier of the packing,written carr P . See Figure 4.

Example 4.3. William Thurston’s original interest in packings began with theinfinite “regular hex packing” in which every circle touches exactly 6 others. Heshowed that the only univalent packing with this combinatorial pattern is theone in which every circle has the same radius. (It remains an open question tocharacterize the non-univalent ones.) See Figure 5.


Figure 6. A portion of the “ball bearing” packing. The carrierhas been drawn in to emphasize the lattice structure.

Example 4.4. Another useful infinite packing is the “ball bearing packing”named by Tomasz Dubejko and Ken Stephenson [DS95b]. The underlying tri-angulation is created from a lattice, and the original lattice structure is stillapparent in the resulting packing. Consequently, the carrier of the packing canbe decomposed into small squares. Moreover, there is a natural refinement ofthe triangulation and carrier created by replacing each square with four copiesof the original. See Figure 6.

4.2. Packings and Maps. The connection between circle packings and func-tion theory arises from the investigation of maps between the carriers of two

different packings for the same abstract complex. That is, suppose P and P areboth Euclidean circle packings for the same underlying complex K. Then every

face in K is realized as both a Euclidean triangle T in carr P and a triangle T

in carr P . It is easy now to construct an affine map between triangles T and T .If we translate one vertex of each to the origin, then the two edges meeting atthe origin form a basis for R

2 and can be mapped one onto the other by a linearmap.

Thus the entire carrier of P can be mapped onto the carrier of P by a piecewiseaffine map defined triangle by triangle. Notice that the individual triangle mapsagree on adjacent edges, so the complete map is continuous. Circle packing mapsconstructed in this way are called discrete conformal maps. See Figure 7.

4.3. The Rodin-Sullivan Theorem. Recall from Section 2.3, that affine mapsare quasiconformal. The dilation on each triangle will be constant and dependonly by the difference between corresponding angles. If there are only finitelymany circles in the packings, the dilatation of a discrete conformal map willbe finite and depend only on the maximal difference in corresponding anglesbetween triangles in the two carriers.


Figure 7. Two circle packings with the same underlying trian-gulation. The carrier for each is indicated and one pair of corre-sponding triangles are shaded. Each triangle in the carrier on theleft can be mapped via an affine map to its corresponding trianglein the carrier on the right.

At this point in our story, we come to Burt Rodin and Dennis Sullivan’s RingLemma, the first connection between the analytic properties of discrete conformalmaps and the combinatorial properties of packings [RS87].

Ring Lemma. In a univalent packing, there is a lower bound Cn on the ratio of

the radius of any interior circle to the radius of any of its neighbors. This bound

depends only on the degree n (the number of neighbors) of the circle.

The sharp value of the bound Cn was determined by Dov Aharonov [Aha97].

Lemma 4.5. If an is the Fibonacci sequence, then

Cn =1

a2

n−2+ a

2

n−1− 1

.

Moreover,Cn

Cn+1

converges to the square of the golden ratio.

The Ring Lemma thus connects a purely combinatorial property of the packing(the degree) with a geometric property of the packing (the ratio of the radii ofadjacent circles). This geometric constraint on the circles implies angles in thecarrier must be bounded away from 0 and π. Hence there is a uniform boundon the difference between corresponding angles in the carriers of two packingswith the same underlying triangulation. Consequently, the associated discreteconformal map is quasiconformal with a bound on the dilatation determinedonly the degree. In this way, a combinatorial property of the triangulation leadsdirectly to an analytic property of the associated discrete conformal maps.

In 1985, William Thurston conjectured the relationships between the combina-torics, geometry, and mapping properties of packings run much deeper [Thu85].


Figure 8. A cross-shaped packing (left) which has been re-packedin the unit disc (right). Since both packings share the same under-lying triangulation, there is discrete conformal map between themwhich approximates the classical Riemann map.

Suppose Ω ( C is a bounded simply connected region and p, q ∈ Ω, p 6= q. TheRiemann Mapping Theorem implies there is a unique conformal map f : Ω → D

with f(p) = 0 and f(q) > 0.

Now suppose Pn is a sequence of packings in Ω with mesh (radius of the largestcircle) decreasing to 0 and carrPn → Ω as n → ∞. Let Kn be the underlyingtriangulation of Pn. Paul Koebe [Koe36], E. M. Andreev [And70a,And70b], andWilliam Thurston [Thu] independently proved that any finite, simply connectedCP-complex (such as Kn) can be realized by a packing in D which is “maximal”in the sense that boundary circles are tangent to ∂D. This maximal, or Andreev,packing is unique up to disc automorphisms.

Thus for each Pn ⊂ Ω, there is a maximal packing Pn ⊂ D with the same

underlying triangulation Kn as Pn. Moreover, we can normalize Pn so thatif Cp and Cq are the nearest circles in Pn to p and q, respectively, then the

corresponding circles Cp and Cq in Pn are centered at 0 and on the positive realaxis, respectively.

Since Pn and Pn share the same underlying triangulation, there is a discreteconformal map

fn : carr Pn → carr Pn.

William Thurston conjectured that fn → f locally uniformly on Ω as n →∞ [Thu85]. This was quickly proven by Burt Rodin and Dennis Sullivan [RS87].See Figure 8.

Rodin-Sullivan Theorem. The discrete conformal maps described above con-

verge locally uniformly to the conformal map f : Ω → D with f(p) = 0 and

f(q) > 0.

Recall that if the degree of Kn is uniformly bounded for all n (Thurston’soriginal conjecture was for packings with degree 6), then the Ring Lemma implies


each fn will be K-quasiconformal, with K independent of n. It remains to showthat the dilatation of fn must actually decrease to 1 as n → ∞. This followsfrom the uniqueness of infinite packings.

Theorem 4.6. Every infinite, simply connected CP-complex has a packing in

either C or D. This packing is unique up to conformal automorphisms.

Various versions of Theorem 4.6 have been proven. Thurston’s original proofwas only for the regular hex packing of Example 4.3 and relied on deep resultsfrom the theory of hyperbolic 3-manifolds [Thu]. Later improvements by KenStephenson [Ste96], Alan Beardon and Ken Stephenson [BS90], Yves Colin deVerdiere [dV89,dV91], Zheng-Xu He and Burt Rodin [HR93], and Zheng-Xu Heand Oded Schramm [HS96, HS98] utilized probabilistic techniques, variationalprinciples, the Perron method, or elementary topology.

The effect of Theorem 4.6 is to force the dilation of fn to decrease to 1 asn → ∞. Consider a circle C “deep inside” Pn, that is, separated from ∂Ω by agreat many generations of other circles. If C is far enough from the boundary,it can hardly tell if it is part of a finite packing, or the unique infinite one. The

same must be true for the corresponding circle C in Pn ⊂ D. Thus triangles in

carr Pn and carr Pn which are far from the boundary, must be nearly the same(up to scaling, translation, and rotation). In particular, the corresponding anglesmust be nearly the same, and the resulting affine map must be nearly conformal.This is usually stated as the Packing Lemma [Ste96,Ste05].

Packing Lemma. Suppose Kn is a sequence of simply connected CP-complexes

with uniformly bounded degree and having univalent packings Pn in a bounded

simply connected domain Ω. If Pn is any other sequence of univalent packings

for Kn, then the maximum difference between corresponding angles in carr Pn

and carr Pn goes to 0 locally uniformly as n → ∞.

Finally, recall that we assumed the mesh of Pn decreased to 0 as n → ∞;thus on compact subsets of Ω, the number of generations of circles betweenthe compact subset and the boundary must go uniformly to infinity as n → ∞.Consequently, the dilatation of fn will decrease to 1 uniformly on compact subsetsof Ω.

5. Applications

5.1. Image Recognition. In work with Ken Stephenson, we have applied cir-cle packing techniques to two-dimensional image recognition problems. DavidMumford and Eitan Sharon have recently developed a technique for studyingtwo-dimensional shapes (Jordan curves) by means of the Weil-Peterson metric ontheir associated welding homeomorphisms [MS04]. They restrict their attentionto smooth curves which then produce diffeomorphisms of ∂D. The Weil-Petersonmetric on these diffeomorphisms is invariant under Mobius transformations; thusshapes which differ only by scaling or rotation are recognized as being the same.


(0.000,0.000)

(6.283,6.283)

Figure 9. A cross-shaped quasicircle (left) and the graph of theresulting quasisymmetry, parametrized as a map from [0, 2π] onto[0, 2π].

It is relatively easy to extend their program to shapes bounded by quasicirclesand to quasisymmetric maps on ∂D. By packing both the inside Ω and outsideΩ∗ of a quasicircle Γ, then repacking in D and D

∗, respectively, we can creatediscrete analytic functions

fn :Ωn → D

gn :Ω∗n → D

∗,

where Ωn → Ω and Ω∗n → Ω∗.

It is much trickier to compare the boundary values of fn and gn since thepackings in Ω and Ω∗ don’t necessarily match up on the boundary. However, itis possible with careful application of the geometry of quasicircles and a dash atopology to create a map

ϕn : ∂D → ∂D

which is essentially given by gnf−1

n . We then have the following theorem [Wil01a]:

Theorem 5.1. The mappings ϕn converge uniformly to the quasisymmetry ϕ

induced by the quasicircle Γ. Moreover, fn and gn converge locally uniformly to

the Riemann maps f : Ω → D and g : Ω → D∗, respectively.

For example, consider the cross-shaped curve in Figure 9. Creating discreteconformal maps as described above (Recall Figure 8), we can approximate thecorresponding quasisymmetry.

Repeating this procedure for a T-shaped curve and a hand-drawn cross inFigures 10 and 11, the similarities and differences with the straight-sided crossare easy to see.

A more difficult problem is to recover the shape given the map ϕ : ∂D →∂D. The Conformal Welding Theorem guarantees that this is possible, but isno help in actually computing the shape. Again, circle packing comes to the


(0.000,0.000)

(6.283,6.283)

Figure 10. A T-shaped quasicircle (left) and the graph of theresulting quasisymmetry (right).

(0.000,0.000)

(6.283,6.283)

Figure 11. A hand-drawn cross (left) and the graph of the re-sulting quasisymmetry (right). Compare with Figures 9 and 10.

Figure 12. A discrete welding for the map ϕ(eiθ) = ei(θ+ 1

3sin(3θ)).

The circles corresponding to the “seam” in packing (left) areshaded. The packing provides a realization on S

2 of the weld-ing triangulations (middle). The edges along the “seam” form thediscrete welding curve (right).


rescue. Instead of welding D to D∗, we will weld triangulations of discs. For

example, if ϕ is a homeomorphism from the boundary of an triangulation K tothe boundary of K∗, we use ϕ to glue the triangulations together. After a fewminor refinements and adjustments, we attach every boundary edge e of K to its

image ϕ(e). This discrete welding then yields a triangulation K of a sphere.

The welded triangulation K can be realized by a unique circle packing on S2.

The uniqueness of this packing is exactly analogous to the uniqueness of theconformal structure on S

2. The circles must push and pull against each other to

settle in locations compatible with the global pattern provided by K in preciselythe same way that two welded discs settle in locations compatible with the globalconformal structure on S

2. This circle packing provides a geometric realizationof the formerly purely combinatorial welding. In particular, the “seam” betweenthe original triangulations is realized as a polygonal Jordan curve, a discretizedversion of the conformal welding curve.

Notice also that K contains a copy of both K and K∗. Thus we can define

discrete analytic functions from K and K∗ onto their copies in K. This is, ofcourse, analogous to the existence of classical welding maps f and g onto com-

plementary regions of S2. Moreover, because of the way we used ϕ to weld K

together, a version of the welding condition g f−1 = ϕ also holds.

In fact, the discrete version is more than just analogous to the classical case –it converges to it as well. Welding finer and finer triangulations using the samequasisymmetric map produces discrete welding curves that converge uniformly tothe classical conformal welding curve. Moreover, the discrete analytic functionsconverge locally uniformly to the classical conformal welding maps [Wil04].

Discrete Conformal Welding Theorem. Given a quasisymmetric map ϕ :∂D → ∂D, our construction produces discrete analytic functions fn and gnconverging locally uniformly to the conformal welding maps f and g induced by

ϕ. Moreover, the discrete conformal welding curves Γn converge uniformly to the

quasicircle Γ induced by ϕ.

5.2. Radnell-Schippers Quantum Field Theory. Recently David Radnelland Eric Schippers [RS05] have developed a two-dimensional quantum field the-ory based on conformal welding and rigged Teichmuller spaces. Very briefly, oneof fundamental ideas of string theory is that a one-dimensional closed string willsweep out a surface, called its world sheet, as it travels through time. As astring breaks apart and rejoins with itself, it alters the topology of the worldsheet. See Figure 13. Dennis Sullivan and Moira Chas have in this manner de-scribed the topology of all world sheets in terms of the splitting and joining ofstrings [Sul01].

While the topology of the world sheet captures the splitting and re-joining of astring, it is the conformal structure of the world sheet that captures features suchas the relative size of the string and length of time between splittings and joinings.Thus for many computations it is necessary to consider all possible conformalstructures on all possible surfaces. The Universal Teichmuller space contains


Figure 13. A depiction of a string traveling through time. Asthe string breaks into two pieces and then rejoins, a topologicalhandle is created.

the Teichmuller spaces of all Riemann surfaces and as such has recently gainedthe attention of physicists as a possible setting for string theory computations[Pek94,Pek95].

Two common models for the Universal Teichmuller space are the space ofnormalized quasicircles and the space of normalized quasisymmetries. The pro-cess of conformal welding described in Section 3.2 provides the mechanism forswitching between the two models [Leh87,Krz95]. Our method of discrete con-formal welding described above provides the means for actually computing thiscorrespondence as well [Wil01a,Wil04].

In the Radnell-Schippers model of quantum field theory, the ends of the worldsheets are parametrized (“rigged”) by quasisymmetric maps. The interactionbetween two strings then corresponds to the welding of the two worldsheets viathe rigging [RS05]. These operations can be carried out using circle packings toapproximate the world sheets. The packable surfaces are dense [Bro86, Bro92,Bro96,BS92,BS93,Wil03] in the moduli space of all surfaces, so nothing is lostin this approach, while much is gained by the ability to actually compute thenew welded surface.

5.3. Circle Packing Measurable Riemann Mapping Theorem. Recallthat the distortion of a quasiconformal map f is described by its complex di-latation µ, defined by the Beltrami equation

(5.1) ∂f = µ ∂f.

The classical Measurable Riemann Mapping Theorem asserts that given a Bel-trami differential µ on a simply connected domain Ω ( C, there is a correspond-ing quasiconformal map f

µ from Ω to the unit disc D having µ as its complexdilatation. If f

µ is normalized to send two points p, q ∈ Ω, p 6= q, to 0 andthe positive real axis, respectively, then f

µ is unique [LV73,Leh87,GL00].. Theoriginal Riemann Mapping Theorem follows from the special case µ = 0.

Circle packings have been used previously by Zheng-Xu He [He90] to solveBeltrami differential equations, but they appear indirectly. By applying ourdiscrete conformal welding technique, however, we can create quasiconformalmaps directly from their complex dilatation.


Given a Beltrami differential µ on a bounded simply connected region Ω ( C,we pack Ω with a “ball bearing” packing. See Figure 6. The carrier divides Ωinto small squares. We approximate µ by a constant function on each square.Recall from Section 2.3 that a map with constant dilatation is affine.

In work with Roger Barnard, we showed that the conformal structure on anycompact torus can be transformed into any other by cutting it open appropriatelyand welding it back together [BW02]. However, the conformal structures ofcompact tori can also be distorted by affine maps. Thus our work on weldingtori provides the mechanism for creating the effect of affine maps.

By refining our ball-bearing packing and performing a discrete conformal weld-ing on each of the small squares in Ω, we can create a normalized discretequasiconformal map fn whose dilatation is approximately equal to µ on eachsquare [Wil].

Circle Packing Measureable Riemann Mapping Theorem. As the pack-

ings are refined, the discrete quasiconformal maps fn converge to the similarly

normalized quasiconformal map fµ : Ω → D with dilation µ.

References

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804.

G. Brock Williams Address: Department of Mathematics, Texas Tech University,

Lubbock, Texas 79409

E-mail: [email protected]: http://www.math.ttu.edu/∼williams

List of registered participants for the workshop(excluding unregistered research scholars, Department of

Mathematics, IIT Madras)

Australia:T-W. MaSchool of Mathematics and StatisticsThe University of Western AustraliaNedlands, W.A., [email protected]

Finland:Peter A. HastoDepartment of Mathematical SciencesP.O. Box 3000, FI-90014University of Oulu, [email protected]

Ilkka HolopainenDept. of Mathematics and StatisticsP.O. Box 68, FI-00014University of Helsinki, [email protected]

Riku KlenDepartment of MathematicsUniversity of TurkuFI-20014 Turku, [email protected]

Henri LindenDept. of Mathematics and StatisticsP.O. Box 68, FI-00014University of Helsinki, [email protected]

Olli MartioDept. of Mathematics and StatisticsP.O. Box 68, FI-00014University of Helsinki, [email protected]

Raimo NakkiDept. of Mathematics and StatisticsP.O. Box 35 (MaD)FIN-40014University of Jyvaskyla, [email protected]

Istvan PrauseDept. of Mathematics and StatisticsP.O. Box 68, FI-00014University of Helsinki, [email protected]

Antti RasilaHelsinki University of TechnologyInstitute of MathematicsP.O. Box 1100, FIN-02015 HUT, [email protected]

Matti VuorinenDepartment of MathematicsUniversity of TurkuFI-20014 Turku, [email protected]

India:Javid AliDepartment of MathematicsAligarh Muslim UniversityAligarh–202 002, [email protected]

Meena S. AtakMaharashtra Academy of EngineeringAlandi–PunePin–412 105meena [email protected]

350 List of Participants IWQCMA05

A. AvudainayagamDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

R BalasubramanianDepartment of MathematicsThe Institute of Mathematical SciencesC.I.T. Campus, TaramaniChennai – 600 [email protected]

S.S. BhoosnurmathDepartment of MathematicsKarnatak UniversityPavate Nagar, Dharwad – 580 003Karnataka State

Bappaditya BhowmikDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

BhuvanaHindustan Engineering CollegePadur, Kanchipuran [email protected]

K. ChandrasekhranR-3 Sivakami ApartmentsSomasundharam StreetMuthuvel Nagar, East [email protected]

S.A. ChoudumDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

Sukhjit Singh DhaliwalDepartment of MathematicsSant Longowal Inst. of Engg. & Tech.Longowal-148106 (Punjab)sukhjit [email protected]

GeethaDepartment of MathematicsDheivanai Ammal College for WomenVillupuram – 605 602Tamil Nadu

K.R. Karthikeyan24, Dhanalakshmi Ammal StreetReddiar Garden, Kamaraj NagarAvadi, Chennai–600 071kr [email protected]

S. KarthikeyanDepartment of MathematicsSona College of TechnologySona NagarThiagarajar Polytechnic College RoadSalem – 636 005, Tamil [email protected]

S.M. KhairnarDepartment of MathematicsAnuradha Engineering College,Chikhli–443201, Dist. Buldana, (M.S)[email protected]

KokilaDepartment of MathematicsDheivanai Ammal College for WomenVillupuram – 605 602Tamil Nadu

V. Lakshmi52, Choolai high roadchoolai, chennai – 600 112Tamil [email protected]


J. Lourthu MaryDepartment of MathematicsMadras Institute TechnologyAnna UniversityChennai – 600 044

N. MarikkannanDepartment of Applied MathematicsSri Venkateswara College of Engg.Pennalur, Sripeumbudur – 602 [email protected]

A.K. MisraDepartment of MathematicsBerhampur UniversityBhanja BiharBerhampur, [email protected]

Debasisha MishraDepartment of MathematicsUtkal University, Vani ViharBhubaneswar – 751004kapa [email protected]

Sumit MohantyC/O Jagannath PatelDepartment of MathematicsUtkal University, Vani ViharBhubaneswar – 751004sumit [email protected]

S. NandaNorth Orissa UniversitySriram Chandra Vihar, TakatpurMayurbhanjBaripada – [email protected]

U.H. NaikDepartment of MathematicsWillingdon CollegeSANGLI – [email protected]

Tarakanta NayakDepartment of MathematicsIIT Guwahati, North GuwahatiGuwahati – [email protected]

Sanjay kumar PantDeen Dayal Upadhyaya college(DU)Shivaji Marg, KarampuraNew Delhi – 110 [email protected]

R. ParvathamRamanujan Institute (RIASM)University of MadrasChennai – 600 [email protected]

J. PatelDepartment of MathematicsUtkal University, Vani ViharBhubaneswar–[email protected]

Santosh Kumar PattanayakDepartment of MathematicsChennai Mathematical InstitutePlot H1, SIPCOT IT ParkPadur PO, Siruseri – 603 [email protected]

S. PonnusamyDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

D.J. PrabhakaranDept. of Appl. Sci. & HumanitiesM.I.T. Campus, Anna UniversityChennai – 600 [email protected]


Z. RebekalDr. M.G.R. Deemed UniversityNo. 122, Saniyasipuram2nd street, MedawalkamKilpauk, Chennai – 600 010

RathigaDepartment of MathematicsAarupadai Veedu Inst. of [email protected]

Satyajit RoyDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

Pravati SahooDepartment of MathematicsMahila Maha Vidyalaya (MMV)Banaras Hindu UniversityVaranasi – [email protected]

Swadesh Kumar SahooDepartment of MathematicsIIT MadrasChennai - 600 036swadesh [email protected]

N.D. SangleAnnasaheb Dange Collegeof Engg. & Tech.Ashta, Dist. Sangli – 416 301Maharastranavneet [email protected]

R. Sathya PriyaR.M.K. Engineering CollegeKavarai PattaiThiruvalluvar – 601 206

S. Selvaganesh1/D Jawaharlal Street, MadupetGudiyattam – 632602Vellore District, Tamil Naduselva [email protected]

C. S. SeshadriChennai Mathematical InstitutePlot H1, SIPCOT IT ParkPadur PO, Siruseri – 603 [email protected]

R. SivakumarDepartment of MathematicsSona College of TechnologySona NagarThiagarajar Polytechnic College RoadSalem – 636 005, Tamil [email protected]

S. Sivaprasad KumarDepartment of Applied MathematicsDelhi College of EngineeringBawana Road, New Delhi – 110 [email protected]

C.M. SubalakshmiDr. M.G.R. Deemed UniversityNo. 23, Bashyam Reddy 2nd streetOoteri, Chennai – 600 012

K. SubathraDepartment of MathematicsRajalakshmi Engg.CollegeRajalakshmi Nagar, Thaudalamk [email protected]

M. SuganthiDept. of Math. and Computer Appl.PSG College of TechnologyCoimbatore – 641004msugan [email protected]


S. SundarDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

Jagmohan TantiDepartment of MathematicsChennai Mathematical InstitutePlot H1, SIPCOT IT ParkPadur PO, Siruseri – 603 [email protected]

G. ThirupathiAdhiyamaan College of EngineeringHosur–635109,Tamil Nadugt [email protected]

E. UmamaheswariDr. M.G.D. Educational & ResearchInst.No. 23, Venkatachala Mudali streetVepery – 600 007

VanithaDepartment of MathematicsDheivanai Ammal College for WomenVillupuram – 605 602Tamil Nadu

Allu VasudevaraoDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

P. VasundhraDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

P. VeeramaniDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

Murali K. VemuriDepartment of MathematicsChennai Mathematical InstitutePlot H1, SIPCOT IT ParkPadur PO, Siruseri – 603 [email protected]

V. VetrivelDepartment of MathematicsIIT MadrasChennai - 600 [email protected]

Iran:R. AghalaryDepartment of MathematicsUniversity of UrmiaUrmia, [email protected]

Saeid ShamsDepartment of MathematicsUniversity of UrmiaUrmia, [email protected]


Japan:Toshiyuki SugawaDepartment of MathematicsGraduate School of ScienceHiroshima University1-3-1 Kagamiyama, Higashi-Hiroshima739-8526 [email protected]

Hiroshi YanagiharaDepartment of Applied ScienceYamaguchi UniversityFaculty of EngineeringTokiwadai Ube 755 – [email protected]

Korea:Jae Ho ChoiDepartment of Mathematics EducationDaegu National University of Educa-tion1797-6 Daemyong 2 dong, NamguDaegu 705-715, [email protected]

Mexico:R. Michael Porter K.Departamento de MatematicasCINVESTAV-I.P.N.Apdo. Postal 14-74007000 Mexico, D.F. [email protected]

Nepal:Ajaya SinghCentral Department of MathematicsTribhuvan UniversityKirtipur, KathmanduNepalajayas [email protected]

C.M. PokhrelDepartment of MathematicsNepal Engineering CollegeG.P.O. Box 10210Kathmandu, [email protected]

USA:Om P. AhujaDepartment of MathematicsKent State UniversityOhio, [email protected]

Roger W. BarnardDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409, [email protected]

Phillip BrownTexas A&M University GalvestonPO Box 1675, GalvestonTexas 77553 -1675, [email protected]

J. Lee BumpusDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409, U.S.Ajlee [email protected]

Atul DixitDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409, [email protected]

D. FreemanDepartment of Mathematical SciencesUniversity of CincinnatiCincinnati, OH 45221-0025, [email protected]


David A. HerronGraduate Program DirectorDepartment of Mathematical SciencesUniversity of CincinnatiCincinnati, Ohio 45221-0025, [email protected]

Casey HumeDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409, [email protected]

William MaDepartment of MathematicsPennsylvania College of Technology22 Hillview Avenue, Williamsport, [email protected]

S.S. MillerDepartment of MathematicsState University of New YorkBrockport, NY 14420, [email protected]

David MindaDepartment of Mathematical SciencesUniversity of CincinnatiCincinnati, OH 45221-0025, U.S.AE-mail: [email protected]

Eric MurphyU.S. Air Force6611 Comet CircleApt no. 302Springfield, VA 22150, [email protected]

Kent PearceDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409, [email protected]

Len RuthDepartment of Mathematical SciencesUniversity of CincinnatiCincinnati, OH 45221-0025, [email protected]

Alex WilliamsDept of Mathematics and StatisticsTexas Tech UniversityLubbock, TX 79409alejandros [email protected]

Brock WilliamsDept. of Mathematics and StatisticsTexas Tech UniversityLubbock, Texas 79409, [email protected]


List of unregistered participants(Research scholars, Department of Mathematics, IIT Madras)

• A. Anthony Eldred• J. Anuradha• A. Chandrashekaran• R. Indhumathi• Nachiketa Mishra• V. Murugan• Param Jeet• H. Ramesh• Jajati Keshori Sahoo• V. Sankaraj• E. Satyanarayana• S. Suresh Kumar• Manoj Yadav

Ponnusamy Sugawa, Vuorinen. (Eds.) Quasiconformal Mappings and Their Applications. IWQCMA05 (Draft, Madras, 2006)(ISBN 8173198071)(358s)_MCc

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