Pseudo Harmonic Morphisms on Riemannian Polyhedra · 2018-08-03 · vide several examples: smooth Riemannian manifolds, Riemannian orbit spaces, normal analytic spaces, Thom spaces

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Pseudo Harmonic Morphisms on RiemannianPolyhedra

M. A. Aprodu, T. Bouziane

Abstract

The aim of this paper is to extend the notion of pseudo harmonic morphism(introduced by Loubeau [13]) to the case when the source manifold is an admissibleRiemannian polyhedron. We define these maps to be harmonic inthe sense of Eells-Fuglede [7] and pseudo-horizontally weakly conformal in our sense (see Section3). We characterize them by means of germs of harmonic functions on the sourcepolyhedron, in sense of Korevaar-Schoen [11], and germs of holomorphic functionson the Kahler target manifold.

Keywords and phrases: Harmonic maps, Harmonic morphisms, Riemannian polyhedra, PHWCmaps, PHM maps, Kahler manifolds.

1 Introduction.

”Harmonicity” is a topic which is situated between geometryand analysis. For instance,Fuglede [8] and Ishihara [10], independently, proved that harmonic morphisms betweensmooth Riemannian manifolds (maps which pull back germs of harmonic functions togerms of harmonic functions) are precisely harmonic maps (analytic property) which arehorizontally weakly conformal (geometric property). A natural question arises: is thereany equivalent notion if the target manifolds are Hermitianor Kahler? If yes, can wecharacterize geometrically this notion? Loubeau, in [13],gave complete answers to thesequestions and named the maps ”Pseudo harmonic morphism”.

In [11], Korevaar and Schoen extended the theory of harmonicmaps between smoothRiemannian manifolds to the case of maps between certain singular spaces: for example

M. A. Aprodu: Department of Mathematics, University of Galati, Domneasca Str. 47, RO-6200,Galati, Romania.e-mail: [email protected]. Bouziane: The Abdus Salam International Center for Theoretical Physics, strada Costiera 11,34014 Trieste , Italy.e-mail: [email protected]

Mathematics Subject Classification (2000): 58E20, 53C43, 53C55, 32Q15.

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admissible Riemannian polyhedra. The Riemannian polyhedra are both very interestingexamples of the ”geometric habitat” of the harmonicity (being harmonic spaces) and pro-vide several examples: smooth Riemannian manifolds, Riemannian orbit spaces, normalanalytic spaces, Thom spaces etc. Later, Eells and Fuglede in [7], expanded the notionof harmonic morphisms to the case of Riemannian polyhedra. But, to give the samecharacterization for harmonic morphisms between Riemannian polyhedra as Fuglede andIshihara did in the smooth case, they had to pay a price: the target had to be a smoothRiemannian manifold. Also, many of the properties found forthe harmonic maps andharmonic morphisms in the smooth case could be recovered when consider as domainand target Riemannian polyhedra.

Remaining in the same spirt of ideas, the aim of this paper is to extend pseudo har-monic morphisms to the case when the domain is an admissible Riemannian polyhedronand the target a Kahler manifold and to characterize them (as it was done in the smoothcase) by ”geometric criteria” and ”analytic criteria”. It turns out, because of the absenceof global differential calculus on singular spaces, that itis not easy to find a good defini-tion of the pseudo harmonic morphisms on Riemannian polyhedra generalizing in a nat-ural way the smooth case. Another difficulty, compare with Loubeau’s results, is to finda geometric condition which characterize pseudo harmonic morphisms on Riemannianpolyhedra, knowing that we can not talk about horizontal vectors for example. A thirddifficulty is in the use of germs of harmonic functions in the sense of Korevaar-Schoen,as the analytic aspect of our construction.

The outline of the paper is as follows. In Section 2 we recall some results on Rieman-nian polyhedra, harmonic maps and morphisms between Riemannian polyhedra. Section3 is devoted to the (alternative) geometric characterization of the pseudo harmonic mor-phisms defined on Riemannian polyhedra named (also) ” pseudo-horizontally weak con-formality ”. We show that this geometric property is preserved by the holomorphy. InSection 4, we introduce the notion of pseudo harmonic morphisms from a Riemannianpolyhedra to a Kahler manifold and characterize them as maps which pull back germsof holomorphic functions on target manifold to germs of harmonic functions on the Rie-mannian polyhedron (Theorem 4.2). We also state a lifting property for pseudo harmonicmorphisms (Proposition 4.7).

Finally, in Section 5, applying Proposition 4.7, we give some interesting examples.

2 Preliminaries.

This section is devoted to some basic notions and known results which will be used in thenext sections.

2.1 Riemannian polyhedra.

2.1.1 Riemannian admissible complexes ([3], [4], [5], [6],[16]).

LetK be a locally finite simplicial complex, endowed with a piecewise smooth Rieman-nian metricg ( i.e. g is a family of smooth Riemannian metricsg∆ on simplices∆ of K,such that the restriction(g∆)|∆′ = g∆′, for any simplices∆′ and∆ with ∆′ ⊂ ∆).

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LetK be a finite dimensional simplicial complex which is connected locally finite. Amapf from [a, b] toK is called a broken geodesic if there is a subdivisiona = t0 < t1 <... < tp+1 = b, such thatf([ti, ti+1]) is contained in some cell and the restriction off to[ti, ti+1] is a geodesic inside that cell. Then define the length of the broken geodesic mapf to be:

L(f) =

p∑

i=0

d(f(ti), f(ti+1)).

The length inside each cell being measured with respect to its metric.Then, defined(x, y), for every two pointsx, y in K, to be the lower bound of the

lengths of broken geodesics fromx to y. d is a pseudo-distance.If K is connected and locally finite, then(K, d) is a length space and hence a geodesic

space (i.e. a metric space where every two points are connected by a curve with lengthequal to the distance between them ) if complete.

A l-simplex inK is called aboundary simplexif it is adjacent to exactly onel + 1simplex. The complexK is calledboundarylessif there are no boundary simplices inK.

The (open)starof an open simplex∆o (i.e. the topological interior of∆ or the pointsof ∆ not belonging to any sub-face of∆; if ∆ is point then∆o = ∆) of K is defined as:

st(∆o) =⋃

{∆oi : ∆i is simplex ofK with ∆i ⊃ ∆}.

The starst(p) of pointp is defined as the star of itscarrier, the unique open simplex∆o

containingp. Every star is path connected and contains the star of its points. In particularK is locally path connected. The closure of any star is sub-complex.

We say that the complexK is admissible, if it is dimensionally homogeneous and forevery connected open subsetU of K, the open setU \ {U ∩ { the(k − 2)− skeleton}}is connected, wherek is the dimension ofK (i.e.K is (n− 1)-chainable).

Let x ∈ K a vertex ofK so thatx is in thel-simplex∆l. We view∆l as an affinesimplex inRl, that is∆l =

⋂li=0Hi, whereH0, H1, ..., Hl are closed half spaces in gen-

eral position, and we suppose thatx is in the topological interior ofH0. The Riemannianmetric g∆l

is the restriction to∆l of a smooth Riemannian metric defined in an openneighborhoodV of ∆l in R

l. The intersectionTx∆l =⋂li=1Hi ⊂ TxV is a cone with

apex0 ∈ TxV , andg∆l(x) turns it into an Euclidean cone. Let∆m ⊂ ∆l (m < l) be

another simplex adjacent tox. Then, the face ofTx∆l corresponding to∆m is isomorphicto Tx∆m and we viewTx∆m as a subset ofTx∆l.

SetTxK =⋃

∆i∋xTx∆i and we call it thetangent coneofK atx. LetSx∆l denote the

subset of all unit vectors inTx∆l and setSx = SxK =⋃

∆i∋xSx∆i. The setSx is called

the link of x in K. If ∆l is a simplex adjacent tox, theng∆l(x) defines a Riemannian

metric on the(l − 1)-simplexSx∆l. The familygx of riemannian metricsg∆l(x) turns

Sx∆l into a simplicial complex with a piecewise smooth Riemannian metric such that thesimplices are spherical.

We call an admissible connected locally finite simplicial complex, endowed with apiecewise smooth Riemannian metric, anadmissible Riemannian complex.

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2.1.2 Riemannian polyhedron [7].

We mean bypolyhedrona connected locally compact separable Hausdorff spaceX forwhich there exists a simplicial complexK and homeomorphismθ : K → X. Anysuch pair(K, θ) is called atriangulationof X. The complexK is necessarily countableand locally finite (cf. [15] page 120) and the spaceX is path connected and locallycontractible. ThedimensionofX is by definition the dimension ofK and it is independentof the triangulation.

A sub-polyhedronof a polyhedronX with given triangulation(K, θ), is the polyhe-dronX ′ ⊂ X having as a triangulation(K ′, θ|K ′) whereK ′ is a subcomplex ofK (i.e.K ′

is the complex whose vertices and simplexes are some of thoseof K).If X is a polyhedron with specified triangulation(K, θ), we shall speak of vertices,

simplexes,i-skeletons or stars ofX respectively of a space of links or tangent cones ofX as the image underθ of vertices, simplexes,i-skeletons or stars ofK respectively theimage of space of links or tangent cones ofK. Thus our simplexes become compactsubsets ofX and thei−skeletons and stars become sub-polyhedrons ofX.

If for given triangulation(K, θ) of the polyhedronX, the homeomorphismθ is locallybi-lipschitz thenX is saidLip polyhedronandθ Lip homeomorphism.

A null set in a Lip polyhedronX is a setZ ⊂ X such thatZ meets every maximalsimplex∆, relative to a triangulation(K, θ) (hence any) in set whose pre-image underθhasn-dimensional Lebesgue measure0, n = dim∆. Note that’almost everywhere’(a.e.)means everywhere except in some null set.

A Riemannian polyhedronX = (X, g) is defined as a Lip polyhedronX with a spec-ified triangulation(K, θ) such that K is a simplicial complex endowed with a covariantbounded measurable Riemannian metric tensorg, satisfying the ellipticity condition be-low. In fact, suppose thatX has homogeneous dimensionn and choose a measurableriemannian metricg∆ on the open euclideann-simplexθ−1(∆o) of K. In terms of eu-clidean coordinates{x1, ..., xn} of pointsx = θ−1(p), g∆ thus assigns to almost everypoint p ∈ ∆o (or x), ann × n symmetric positive definite matrixg∆ = (g∆ij (x))i,j=1,...,n

with measurable real entries and there is a constantΛ∆ > 0 such that (ellipticity condi-tion):

Λ−2∆

n∑

i=0

(ξi)2 6∑

i,j

g∆ij (x)ξiξj 6 Λ2

∆

n∑

i=0

(ξi)2

for a.e. x ∈ θ−1(∆o) and everyξ = (ξ1, ..., ξn) ∈ Rn. This condition amounts to the

components ofg∆ being bounded and it is independent not only of the choice of theeuclidean frame onθ−1(∆o) but also of the chosen triangulation.

For simplicity of statements we shall sometimes require that, relative to a fixed trian-gulation(K, θ) of Riemannian polyhedronX (uniform ellipticity condition),

Λ := sup{Λ∆ : ∆ is simplex ofX} <∞.

A Riemannian polyhedronX is said to beadmissibleif for a fixed triangulation(K, θ)(hence any) the Riemannian simplicial complexK is admissible.

We underline that (for simplicity) the given definition of a Riemannian polyhedron(X, g) contains already the fact (because of the definition above ofthe Riemannian ad-missible complex) that the metricg is continuousrelative to some (hence any) triangula-tion (i.e. for every maximal simplex∆ the metricg∆ is continuous up to the boundary).

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This fact is sometimes in the literature omitted. The polyhedron is said to besimplex-wise smoothif relative to some triangulation(K, θ) (and hence any), the complexK issimplexwise smooth. Both continuity and simplexwise smoothness are preserved undersubdivision.

In the case of a general bounded measurable Riemannian metric g on X, we oftenconsider, in addition tog, the euclidean Riemannian metricge on the Lip polyhedronX with a specified triangulation(K, θ). For each simplex∆, ge∆ is defined in terms ofeuclidean frame onθ−1(∆o) as above by unit matrix(δij)i,j. Thusge is by no meanscovariantly defined and should be regarded as a mere reference metric on the triangulatedpolyhedronX.

Relative to a given triangulation(K, θ) of ann-dimensional Riemannian polyhedron(X, g) (not necessarily admissible), we have onX the distance functione induced bythe euclidean distance on the euclidean spaceV in whichK is affinely Lip embedded.This distancee is not intrinsic but it will play an auxiliary role in definingan equivalentdistancedX as follows:

Let Z denote the collection of all null sets ofX. For given triangulation(K, θ) con-sider the setZK ⊂ Z obtained fromX by removing from each maximal simplex∆ in Xthose points of∆o which are Lebesgue points forg∆. Forx, y ∈ X and anyZ ∈ Z suchthatZ ⊂ ZK we set:

dX(x, y) = supZ ∈ Z

Z ⊃ ZK

infγ

γ(a) = x,γ(b) = y

{LK(γ) :

γ is Lip continuous pathand transversal toZ

},

whereLK(γ) is the length of the pathγ defined as:

LK(γ) =∑

∆⊂X

∫

γ−1(∆o)

√(g∆ij ◦ θ

−1 ◦ γ). γi. γj ,the sum is overall simplexes meetingγ.

It is shown in [7] that the distancedX is intrinsic, in particular it is independent ofthe chosen triangulation and it is equivalent to the euclidean distancee (due to the Lipaffinely and homeomorphically embedding ofX in some euclidean spaceV ).

2.2 Energy of maps

The concept of energy in the case of a map of Riemannian domaininto an arbitrary metricspaceY was defined and investigated by Korevaar and Schoen [11]. Later this conceptwas extended by Eells and Fuglede [7] to the case of map from anadmissible RiemannianpolyhedronX with simplexwise smooth Riemannian metric. Thus, theenergy E(ϕ) ofa mapϕ from X to the spaceY is defined as the limit of suitable approximate energyexpressed in terms of the distance functiondY of Y .

It is shown in [7] that the mapsϕ : X → Y of finite energy are precisely those qua-sicontinuous (i.e. has a continuous restriction to closed sets), whose complements havearbitrarily small capacity, (cf. [7] page 153) whose restriction to each top dimensionalsimplex ofX has finite energy in the sense of Korevaar-Schoen, andE(ϕ) is the sum ofthe energies of these restrictions.

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Now, let (X, g) be an admissiblem-dimensional Riemannian polyhedron with sim-plexwise smooth Riemannian metric. It is not required thatg is continuous across lowerdimensional simplexes. The target(Y, dY ) is an arbitrary metric space.

DenoteL2loc(X, Y ) the space of allµg-measurable (µg the volume measure ofg) maps

ϕ : X → Y having separable essential range and for which the mapdY (ϕ(.), q) ∈L2loc(X, µg) (i.e. locallyµg-squared integrable) for some pointq (hence by triangle in-

equality for any point). Forϕ, ψ ∈ L2loc(X, Y ) define their distanceD(ϕ, ψ) by:

D2(ϕ, ψ) =

∫

X

d2Y (ϕ(x), ψ(y))dµg(x).

Two mapsϕ, ψ ∈ L2loc(X, Y ) are said to beequivalentif D(ϕ, ψ) = 0,(i.e. ϕ(x) = ψ(x)

µg-a.e.). If the spaceX is compact thenD(ϕ, ψ) <∞ andD is a metric onL2loc(X, Y ) =

L2(X, Y ) and complete if the spaceY is complete [11].Theapproximate energy densityof the mapϕ ∈ L2

loc(X, Y ) is defined forǫ > 0 by:

eǫ(ϕ)(x) =

∫

BX(x,ǫ)

d2Y (ϕ(x), ϕ(x′))

ǫm+2dµg(x

′).

The functioneǫ(ϕ) > 0 is locallyµg-integrable.TheenergyE(ϕ) of a mapϕ of classL2

loc(X, Y ) is:

E(ϕ) = supf∈Cc(X,[0,1])

(lim supǫ→0

∫

X

feǫ(ϕ)dµg),

whereCc(X, [0, 1]) denotes the space of continuous functions fromX to the interval[0, 1]with compact support.

A mapϕ : X → Y is said to belocally of finite energy, and we writeϕ ∈ W 1,2loc (X, Y ),

if E(ϕ|U) < ∞ for every relatively compact domainU ⊂ X, or equivalently ifX can becovered by domainsU ⊂ X such thatE(ϕ|U) <∞.

For example (cf. [7] Lemma 4.4), every Lip continuous mapϕ : X → Y is of classW 1,2loc (X, Y ). In the case whenX is compactW 1,2

loc (X, Y ) is denotedW 1,2(X, Y ) thespace of all maps of finite energy.

W 1,2c (X, Y ) denotes the linear subspace ofW 1,2(X, Y ) consisting of all maps of finite

energy of compact support inX.We can show (cf. [7] Theorem 9.1) that a real functionϕ ∈ L2

loc(X) is locally of finiteenergy if and only if there is a functione(ϕ) ∈ L1

loc(X), namedenergy densityof ϕ, suchthat (weak convergence):

limǫ→0

∫

X

feǫ(ϕ)dµg =

∫

X

fe(ϕ)dµg, for eachf ∈ Cc(X).

2.3 Harmonic maps and harmonic morphisms on Riemannian poly-hedra [7].

In this paragraph we shall remind some relevant results which give the relation betweenharmonic morphisms and harmonic maps on Riemannian polyhedra.

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2.3.1 Harmonic maps.

Let (X, g) be an arbitrary admissible Riemannian polyhedron (g just bounded measurablewith local elliptic bounds),dimX = m and(Y, dY ) a metric space .

A continuous mapϕ : X → Y of classW 1,2loc (X, Y ) is said to beharmonicif it is

bi-locally E-minimizing, i.e. X can be covered by relatively compact subdomainsU foreach of which there is an open setV ⊃ ϕ(U) in Y such that

E(ϕ|U) 6 E(ψ|U)

for every continuous mapψ ∈ W 1,2loc (X, Y ), with ψ(U) ⊂ V andψ = ϕ in X\U .

Let (N, h) denote a smooth Riemannian manifold without boundary,dimRN = nandΓkαβ the Christoffel symbols onN . By a weakly harmonic mapϕ : X → N wemean a quasicontinuous map (a map which is continuous on the complement of open setsof arbitrarily small capacity ; in the case of the RiemannianpolyhedronX it is just thecomplement of open subsets of the(m− 2)-skeleton ofX) of classW 1,2

loc (X,N) with thefollowing property:

For any chartη : V → Rn on N and any quasiopen setU ⊂ ϕ−1(V ) of compact

closure in X, the equation∫

U

〈∇λ,∇ϕk〉dµg =

∫

U

λ(Γkαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg,

holds for everyk = 1, ..., n and every bounded functionλ ∈ W 1,20 (U).

It is shown in [7], (Theorem 12.1), that: for a continuous mapϕ ∈ W 1,2loc (X,N) the

following are equivalent:(a)ϕ is harmonic,(b) ϕ is weakly harmonic,(c) ϕ pulls convex functions on open setsV ⊂ N back to subharmonic functions on

ϕ−1(V ).

2.3.2 Harmonic morphisms.

Denote byX andY two Riemannian polyhedra (or any harmonic spaces in the sense ofBrelot; see Chapter 2, [7]).

A continuous mapϕ : X → Y is aharmonic morphismif, for every open setV ⊂ Yand for every harmonic functionv onV , v ◦ ϕ is harmonic onϕ−1(V ).

Letϕ : X → Y be a nonconstant harmonic morphism, then (cf. [7], Theorem 13.1):(i) ϕ likewise pulls germs of superharmonic functions onY back to germs of super-

harmonic functions onX.(ii) If ϕ is surjective and proper then a functionv : V → [−∞,∞] (V open inY ) is

superharmonic [resp. harmonic] if (and only if)v ◦ ϕ is superharmonic [resp. harmonic]onϕ−1(V ).

Let (N, gN) denote an-Riemannian manifold without boundary and suppose that thepolyhedronX is admissible. A continuous mapϕ : X → N of classW 1,2

loc (X,N) iscalledhorizontally weakly conformalif there exist a scalarλ, defineda.e. inX, such that:

〈∇(v ◦ ϕ),∇(w ◦ ϕ)〉 = λ[gN(∇Nv,∇Nw) ◦ ϕ] a.e. in X

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for every pair of functionsv, w ∈ C1(N). Henceforth∇N denote the gradient operator onN and∇ the gradient operator defined a.e. on the domain space(X, g).

The property of horizontally weak conformality is a local one, thus it reads in termsof local coordinates(yα) in N ,

〈∇ϕα,∇ϕβ〉 = λ(gαβN ◦ ϕ) a.e. in X

for α, β = 1, ..., n. Takingα = β, λ is uniquely determined andλ > 0 a.e. inX.Moreover,λ ∈ L1

loc(X) because∇ϕα ∈ L1loc(X). λ is called thedilation of ϕ.

The notion of horizontally weak conformality is intimatelyrelated to the one of har-monic morphisms. For instance we can show (cf. [7], Theorem 13.2) that a continuousmapϕ : X → N of classW 1,2

loc (X,N) is a harmonic morphism if and only ifϕ is hori-zontally weakly conformal, harmonic map and equivalently,there is a scalarλ ∈ L1

loc(X)such that

−

∫

X

〈∇ψ,∇(v ◦ ϕ)〉 =

∫

X

ψλ[(∆Nv) ◦ ϕ]

for everyv ∈ C2(N) andψ ∈ Lipc(X) ( or ψ ∈ W 1,20 (X) ∩ L∞(X)).

In the affirmative case,λ from the last equality equals a.e. the dilation ofϕ (as ahorizontally weakly conformal map).

As a consequence, for a harmonic morphismϕ : X → N , if ψ : N → P is a harmonicmap between smooth Riemannian manifolds without boundary,then the compositionψ◦ϕis a harmonic map.

3 Pseudo-horizontally weakly conformal maps on Rie-mannian polyhedra.

The aim of the present section, is to extend the notion ofpseudo-horizontally weaklyconformalmaps on Riemannian manifolds (see [13]) to Riemannian polyhedra and toestablish their properties. We will use the same terminology as in [13].

Let (X, g) be an admissible Riemannian polyhedron ofdimX = m and(N, JN , gN)a Hermitian manifold ofdimRN = 2n, without boundary.

We denote byHolom(N) = {f : N → C, f local holomorphic function}. In whatfollows, the gradient operator and the inner product in(X, g) are well defined a.e. inXand will be denoted by∇ and〈, 〉 respectively.

Definition 3.1 Let ϕ : X → N be a continuous map of classW 1,2loc (X,N). ϕ is called

pseudo-horizontally weakly conformal(shortening PHWC), if for any pair of local holo-morphic functionsv, w ∈ Holom(N), such thatv = v1 + iv2, w = w1 + iw2, we have:

{〈∇(w1 ◦ ϕ),∇(v1 ◦ ϕ)〉 − 〈∇(w2 ◦ ϕ),∇(v2 ◦ ϕ)〉 = 0 a.e. inX〈∇(w2 ◦ ϕ),∇(v1 ◦ ϕ)〉+ 〈∇(w1 ◦ ϕ),∇(v2 ◦ ϕ)〉 = 0 a.e. inX

(1)

Remark 3.2 Definition 3.1is a local one, hence it is sufficient to check the identities (1)in local complex coordinates(z1, z2, ..., zn) in N . TakingzA = xA + iyA, ∀A = 1, ..., n,

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the relations (1),∀A,B = 1, ..., n, read:

〈∇ϕB1 ,∇ϕA1 〉 − 〈∇ϕB2 ,∇ϕ

A2 〉 = 0 a.e. inX

〈∇ϕB2 ,∇ϕA1 〉+ 〈∇ϕB1 ,∇ϕ

A2 〉 = 0 a.e. inX

(2)

where

ϕA1 := xA ◦ ϕ, ϕA2 := yA ◦ ϕ, ∀A = 1, ..., n,

ϕB1 := xB ◦ ϕ, ϕB2 := yB ◦ ϕ, ∀B = 1, ..., n.

Remark 3.3 Definition 3.1is justified by seeing that if the source manifold is a smoothRiemannian one, without boundary, we obtain exactly the commuting condition betweendϕx ◦ dϕ

∗x andJNϕ(x) (see [13] or [2], [1]), wheredϕ∗

x : Tϕ(x)N → TxX is the adjoint mapof the tangent linear mapdϕx : TxX → Tϕ(x)N , for anyx ∈ X.

The next proposition justifies the use of the term ’horizontally weakly conformal’,indeed we obtain, when the target dimension is one, an equivalence between the hori-zontally weakly conformality m and pseudo-horizontally weakly conformality, as in thesmooth case.

Proposition 3.4 Let ϕ : X → N a horizontally weakly conformal map(see subsec-tion 2.3.2)from a Riemannian admissible polyhedron(X, g) into a Hermitian manifold(N, JN , gN). Thenϕ is pseudo-horizontally weakly conformal. If the complex dimensionofN is equal to one, then the two conditions are equivalent.

Proof: Let ϕ : X → N be a horizontally weakly conformal map from an admissi-ble Riemannian polyhedron(X, g) into a Hermitian manifold(N, JN , gN) of real di-mension2n. Take (zA = xA + iyA)A=1,...,n local complex coordinates inN . Then{ ∂∂x1, ..., ∂

∂xn, ∂∂y1, ..., ∂

∂yn} is a local frame inTN such that

JN( ∂∂xA

) = ∂∂yA

JN( ∂∂yA

) = − ∂∂xA

, ∀A = 1, ..., n.

The horizontally weakly conformal condition reads in the considered frame:

〈∇ϕα,∇ϕβ〉 = λ(gαβN ◦ ϕ) a.e. inX, ∀α, β = 1, ..., 2n.(3)

whereϕα = ξα ◦ ϕ for ξα =

{xα, α = 1, ..., nyα−n, α = n+ 1, ..., 2n

. Explicitly the equalities (3)

are the following:

〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉 = λ[gN(∂∂ξα, ∂∂ξβ

) ◦ ϕ] a.e. inX,

∀α, β = 1, ..., n.〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉 = λ[gN(

∂∂ξα, ∂∂ξβ

) ◦ ϕ] a.e. inX,

∀α, β = n+ 1, ..., 2n.

(4)

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and

〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉 = λ[gN(∂∂ξα, ∂∂ξβ

) ◦ ϕ] a.e. inX,

∀α = 1, ..., n;∀β = n+ 1, ..., 2n.

〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉 = λ[gN(∂∂ξα, ∂∂ξβ

) ◦ ϕ] a.e. inX,

∀α = n+ 1, ..., 2n;∀β = 1, ..., n.

(5)

Read (4) and (5) in terms ofxA andyB, ∀A,B = 1, ..., n:

〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉 = λ[gN(∂∂xA

, ∂∂xB

) ◦ ϕ] a.e. inX,

〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉 = λ[gN(∂∂yA

, ∂∂yB

) ◦ ϕ] a.e. inX.(6)

and

〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉 = λ[gN(∂∂xA

, ∂∂yB

) ◦ ϕ] a.e. inX,

〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉 = λ[gN(∂∂yA

, ∂∂xB

) ◦ ϕ] a.e. inX.(7)

BecauseJN is the complex structure with respect to the hermitian metric gN , we have∀A,B = 1, ..., n:

gN(∂∂xA

, ∂∂xB

) = gN(∂∂yA

, ∂∂yB

),

gN(∂∂xA

, ∂∂yB

) = −gN (∂∂yA

, ∂∂xB

).(8)

Invoking (8), (6) and (7) we conclude thatϕ is pseudo-horizontally weakly conformal.Consider now the case whendimCN = 1 and supposeϕ : X → N is a pseudo-

horizontally weakly conformal map.Let z = x + iy be a local complex chart inN . In terms of this chart the pseudo-

horizontally weakly conformal condition (1) reads:{

〈∇ϕx,∇ϕx〉 − 〈∇ϕy,∇ϕy〉 = 0 a.e. inX〈∇ϕy,∇ϕx〉+ 〈∇ϕx,∇ϕy〉 = 0 a.e. inX

(9)

Remember thatgN( ∂∂x ,∂∂x) = gN(

∂∂y, ∂∂y) 6= 0, so we can define

λ :=〈∇ϕx,∇ϕx〉

gN(∂∂x, ∂∂x) ◦ ϕ

=〈∇ϕy,∇ϕy〉

gN(∂∂y, ∂∂y) ◦ ϕ

, a.e. inX.(10)

From (9) and (10) we get:

〈∇ϕx,∇ϕx〉 = λ[gN(∂∂x, ∂∂x) ◦ ϕ] a.e. inX

〈∇ϕy,∇ϕy〉 = λ[gN(∂∂y, ∂∂y) ◦ ϕ] a.e. inX

〈∇ϕx,∇ϕy〉 = 0 a.e. inX

(11)

which means thatϕ is horizontally weakly conformal. �

10

PHWC maps on Riemannian polyhedra can be characterized thanks to germs of holo-morphic functions on the target Hermitian manifolds as follows:

Proposition 3.5 Letϕ : X → N be a continuous map of classW 1,2loc (X,N). Thenϕ is

pseudo-horizontally weakly conformal if and only if for anylocal holomorphic functionψ : N → C, ψ ◦ ϕ is also pseudo-horizontally weakly conformal.

Proof: Let ϕ : X → N be a continuous map of classW 1,2loc (X,N), andψ : N → C be

any holomorphic function withψ = ψ1 + iψ2.It is obvious (by definition) that ifϕ is a PHWC map thenψ ◦ ϕ is a PHWC map.Conversely, suppose now that for any holomorphic functionψ : N → C, the compo-

sitionψ ◦ ϕ : X → C is a PHWC function. Throughout the proof of Proposition 3.4 wehave seen that this last fact reads:

{〈∇(ψ1 ◦ ϕ),∇(ψ1 ◦ ϕ)〉 − 〈∇(ψ2 ◦ ϕ),∇(ψ2 ◦ ϕ)〉 = 0 a.e. inX〈∇(ψ1 ◦ ϕ),∇(ψ2 ◦ ϕ)〉+ 〈∇(ψ2 ◦ ϕ),∇(ψ1 ◦ ϕ)〉 = 0 a.e. inX

(12)

Then for a fixed local holomorphic chart(zα)α=1,...,n of N , zα = xα + iyα, the equali-ties (12) become:

0 = 〈∇(ψ1 ◦ ϕ),∇(ψ1 ◦ ϕ)〉 − 〈∇(ψ2 ◦ ϕ),∇(ψ2 ◦ ϕ)〉 =n∑

k,l=1

〈(∂kψ1 ◦ ϕ).∇ϕk, (∂lψ1 ◦ ϕ).∇ϕ

l〉−

n∑k,l=1

〈(∂kψ2 ◦ ϕ).∇ϕk, (∂lψ2 ◦ ϕ).∇ϕ

l〉 a.e. inX

and

0 = 〈∇(ψ1 ◦ ϕ),∇(ψ2 ◦ ϕ)〉+ 〈∇(ψ2 ◦ ϕ),∇(ψ1 ◦ ϕ)〉 =n∑

k,l=1

〈(∂kψ1 ◦ ϕ).∇ϕk, (∂lψ2 ◦ ϕ).∇ϕ

l〉+

n∑k,l=1

〈(∂kψ2 ◦ ϕ).∇ϕk, (∂lψ1 ◦ ϕ).∇ϕ

l〉 a.e. inX

(13)

whereϕα = ξα ◦ ϕ for ξα =

{xα, α = 1, ..., nyα−n, α = n+ 1, ..., 2n

.

Taking into account the Cauchy-Riemann equations and afterdoing some computa-tions, (13) becomes:

11

n∑k,l=1

[(∂ψ1

∂xk◦ ϕ)(∂ψ1

∂xl◦ ϕ)− (∂ψ2

∂xk◦ ϕ)(∂ψ2

∂xl◦ ϕ)

]

[〈∇(xk ◦ ϕ),∇(xl ◦ ϕ)〉 − 〈∇(yk ◦ ϕ),∇(yl ◦ ϕ)〉]+

n∑k,l=1

[(∂ψ1

∂xk◦ ϕ)(∂ψ1

∂yl◦ ϕ)− (∂ψ2

∂xk◦ ϕ)(∂ψ2

∂yl◦ ϕ)

]

[〈∇(xk ◦ ϕ),∇(yl ◦ ϕ)〉+ 〈∇(yk ◦ ϕ),∇(xl ◦ ϕ)〉] = 0 a.e. inX

and

n∑k,l=1

[(∂ψ1

∂xk◦ ϕ)(∂ψ2

∂xl◦ ϕ) + (∂ψ2

∂xk◦ ϕ)(∂ψ1

∂xl◦ ϕ)

]

[〈∇(xk ◦ ϕ),∇(xl ◦ ϕ)〉 − 〈∇(yk ◦ ϕ),∇(yl ◦ ϕ)〉]+

n∑k,l=1

[(∂ψ1

∂xk◦ ϕ)(∂ψ2

∂yl◦ ϕ) + (∂ψ2

∂xk◦ ϕ)(∂ψ1

∂yl◦ ϕ)

]

[〈∇(xk ◦ ϕ),∇(yl ◦ ϕ)〉+ 〈∇(yk ◦ ϕ),∇(xl ◦ ϕ)〉] = 0 a.e. inX

(14)

Now, choose particular holomorphic functionsψ’s, for example locallyψ = zk +zl and vary thek, l indices, we obtain all the pseudo-horizontally weakly conformalityconditions of the mapϕ. This ends the proof.

�

The next proposition makes clear the relation between PHWC maps on Riemannianpolyhedra and holomorphic maps on target Hermitian manifolds.

Proposition 3.6 Letϕ : X → N a continuous map of classW 1,2loc (X,N) and(P, JP , gP )

another Hermitian manifold ofdimRP = 2p. Thenϕ is pseudo-horizontally weaklyconformal if and only if for every local holomorphic mapψ : N → P , ψ ◦ ϕ is alsopseudo-horizontally weakly conformal.

Proof: Let ψ : N → P be a local holomorphic map. Choose(zα)α=1,...,p local complexcoordinates inP and denoteψα := zα ◦ ψ, ∀α = 1, ..., p.

Suppose thatϕ is pseudo-horizontally weakly conformal. Then, by definition we have,for every pair of local holomorphic functionsv, w ∈ Holom(N), such thatv = v1 + iv2,w = w1 + iw2,

{〈∇(w1 ◦ ϕ),∇(v1 ◦ ϕ)〉 − 〈∇(w2 ◦ ϕ),∇(v2 ◦ ϕ)〉 = 0 a.e. inX〈∇(w2 ◦ ϕ),∇(v1 ◦ ϕ)〉+ 〈∇(w1 ◦ ϕ),∇(v2 ◦ ϕ)〉 = 0 a.e. inX

(15)

In particular, for every pair of local holomorphic functions on the Hermitian manifoldN ,ψα, ψβ, α, β = 1, ..., p we have:

{〈∇(ψα1 ◦ ϕ),∇(ψβ1 ◦ ϕ)〉 − 〈∇(ψα2 ◦ ϕ),∇(ψβ2 ◦ ϕ)〉 = 0 a.e. inX〈∇(ψα2 ◦ ϕ),∇(ψβ1 ◦ ϕ)〉+ 〈∇(ψα1 ◦ ϕ),∇(ψβ2 ◦ ϕ)〉 = 0 a.e. inX

(16)

12

Or equivalently, if we denotezα = xα + iyα and

{ψα1 = xα ◦ ψψα2 = yα ◦ ψ

, ∀α = 1, ..., p, we

have:

〈∇(xα ◦ (ψ ◦ ϕ)),∇(xβ ◦ (ψ ◦ ϕ))〉−〈∇(yα ◦ (ψ ◦ ϕ)),∇(yβ ◦ (ψ ◦ ϕ))〉 = 0 a.e. inX〈∇(yα ◦ (ψ ◦ ϕ)),∇(xβ ◦ (ψ ◦ ϕ))〉+〈∇(xα ◦ (ψ ◦ ϕ)),∇(yβ ◦ (ψ ◦ ϕ))〉 = 0 a.e. inX

(17)

for everyα, β = 1, ..., p. Which means by definition, thatψ ◦ ϕ is a PHWC map.Conversely, suppose now thatψ ◦ ϕ is pseudo-horizontally weakly conformal for any

local holomorphic mapψ : N → P . Consider a local complex chart(zα)α=1,...,p in P . Sothe mapψα ◦ ϕ, ∀α = 1, ..., p is pseudo-horizontally weakly conformal (in the sense thatwe apply (1) forv = w = zα).

In order to prove thatϕ is a PHWC map we shall use Proposition 3.5. Letu : N → C

denote a local holomorphic function onN . We associate tou a new map:

φu : N → Cp

x 7→ (0, ..., 0, u(x)︸︷︷︸, 0, ..., 0)α

For any local holomorphic chartη : P → Cp, we obtain a mapψu : N → P , ψu =

η−1 ◦ φu such that theα’s coordinate ofψu is exactly the complex functionu : N → C.So we have proved that every local holomorphic functionu onN can be obtained as

a coordinate function of some local holomorphic mapψu : N → P .We have supposed before, that for everyψ : N → P and any local complex chart

(zα)α=1,...,p in P ,

{〈∇(ψα1 ◦ ϕ),∇(ψα1 ◦ ϕ)〉 − 〈∇(ψα2 ◦ ϕ),∇(ψα2 ◦ ϕ)〉 = 0 a.e. in X〈∇(ψα2 ◦ ϕ),∇(ψα1 ◦ ϕ)〉+ 〈∇(ψα1 ◦ ϕ),∇(ψα2 ◦ ϕ)〉 = 0 a.e. in X

In particular, forψ = ψu ( ψα = u), we obtainu ◦ ϕ is pseudo-horizontally weaklyconformal, for anyu ∈ Holom(N). This implies (cf. Proposition 3.5) thatϕ is a pseudo-horizontally weakly conformal map.

�

4 Pseudo harmonic morphism on Riemannian polyhedra.

Similarly to the smooth case, if the target manifold is endowed with a Kahler structureone can enlarge the class of harmonic morphisms on Riemannian polyhedra to the classof pseudo harmonic morphisms.

Let (X, g) denote an admissible Riemannian polyhedron and(N, JN , gN) a Kahlermanifold without boundary.

Definition 4.1 A mapϕ : X → N is called pseudo harmonic morphism(shorteningPHM) if and only ifϕ is a harmonic map ( in the sense of Korevaar-Schoen[11] andEells-Fuglede[7]) and pseudo-horizontally weakly conformal.

13

Now, we will give a characterization of a pseudo harmonic morphism in terms ofthe germs of the holomorphic functions on the target Kahlermanifold, and the harmonicstructure (in sense of Brelot, see ch 2 [7]) on the domain admissible Riemannian polyhe-dron.

Theorem 4.2 A continuous mapϕ : X → N of classW 1,2loc (X,N) is a pseudo harmonic

morphism if and only ifϕ pulls back local complex-valued holomorphic functions onNto harmonic functions onX (i.e. for any holomorphic functionψ : V → C defined on aopen subsetV ofN with ϕ−1(V ) non-empty, the compositionψ ◦ ϕ : ϕ−1(V ) → R

2 isharmonic if and only ifϕ is PHM).

In order to prove the theorem, we shall need the following elementary lemma:

Lemma 4.3 Letϕ ∈ W 1,2loc (X,N) andψ ∈ Holom(N). If ψ = ψ1 + iψ2, thenψ ◦ ϕ ∈

W 1,2loc (X,R

2) and moreoverψj ◦ ϕ ∈ W 1,2loc (X), ∀j = 1, 2.

Proof: Let ψ : N → C be a holomorphic function andK ⊂ N denote a compact subsetin N . We say thatψ is uniformly Lipschitz inK if there is a scalarλ (depending onK),such that||ψ(p)−ψ(q)|| 6 λdN(p, q), ∀p, q ∈ K, where||.|| denotes the usually norm inC anddN is the associated distance to the Riemannian metricgN onN . Soψ is uniformlyLipschitz onK.

We takeU a quasiopen set onX which is relatively compact.ψ|ϕ(U) is uniformlyLipschitz.

Now, following a result of Eells-Fuglede (see Corollary 9.1, p.158, [7])

EU(ψ ◦ ϕ) 6 λ2ϕ(U)

EU(ϕ).

It suffices to coverX by a countable quasiopen setsU , such thatEU(ϕ) <∞ (such thingis possible becauseϕ ∈ W 1,2

loc (X,N)). We conclude thatψ ◦ ϕ ∈ W 1,2loc (X,R

2).Forψj ◦ ϕ, ∀j = 1, 2, just remark thatψj, ∀j = 1, 2 are locally Lipschitz, so by the

same argument used for proving thatψ ◦ ϕ ∈ W 1,2loc (X,R

2), we haveψj ◦ ϕ ∈ W 1,2loc (X),

∀j = 1, 2. �

Remark 4.4 Let f : N → C be a local holomorphic function defined on a complexmanifoldN . Take(z1, ..., zn) a local holomorphic coordinates inN and denotezj =xj + yj, ∀j = 1, ..., n. If the manifoldN is Kahler and if we writef = f 1 + if 2, than wehave the following equalities (”symmetries”):

∂2f j

∂xA∂yB=

∂2f j

∂xB∂yA, ∀j = 1, 2; ∀A,B = 1, ..., n.

Proof of Theorem 4.2:” ⇒ ”:Letϕ : X → N be a continuous map of classW 1,2

loc (X,N). Suppose thatϕ is pseudo-harmonic morphism and letψ : N → C be any local holomorphic function, withψ =ψ1 + iψ2. Remark thatψ ◦ ϕ is a continuous map ( as a composite of two continuousmaps). Moreover, by Lemma 4.3,ψ ◦ ϕ ∈ W 1,2

loc (X,R2).

14

In order to prove thatψ ◦ ϕ is harmonic, (cf. subsection 2.3.1) it suffices to show thatψ ◦ ϕ is weakly harmonic (as a map with values inR2).

The fact thatψ◦ϕ is weakly harmonic reads in the unique (conformal) chartη : R2 →R

2 and for any quasiopen setU ⊂ ϕ−1(R2) of compact closure inX,∫U

〈∇f,∇(ψi ◦ ϕ)〉dµg =

=∫U

f.(R2

Γiαβ ◦ ϕ)〈∇(ψα ◦ ϕ),∇(ψβ ◦ ϕ)〉dµg,(18)

for i = 1, 2 and every bounded functionf ∈ W 1,2c (U).

In the case ofR2 we know thatR2

Γiαβ ≡ 0, for anyα, β = 1, 2, so the equations (18)become: ∫

U

〈∇f,∇(ψi ◦ ϕ)〉dµg = 0, ∀i = 1, 2.(19)

Let us prove now (19).Take a real chart(x1, ..., xn, y1, ..., yn) in N , such that the complex associated chart

(zj = xj + iyj)j=1,n is holomorphic. Then for any chart domainV ⊂ N and quasiopensetU ⊂ ϕ−1(V ) of compact closure inX, and for any functionsv ∈ C2(V ) andf ∈W 1,2c (U) ∩ L∞(U) we have, (cf. [7] Remark 9.7):

∫

U

〈∇f,∇(v ◦ ϕ)〉dµg =

∫

U

〈[(∂αv) ◦ ϕ]∇f,∇ϕα〉dµg,(20)

where∂α denotes theα’s partial derivative.By partial integration,

∫U

〈∇f,∇(v ◦ ϕ)〉dµg =∫U

〈∇(f.[(∂αv) ◦ ϕ]),∇ϕα〉dµg−

∫U

f.[(∂α∂βv) ◦ ϕ]〈∇ϕα,∇ϕβ〉dµg

(21)

Recall that in the local coordinates(x1, ..., xn, y1, ..., yn) in N , we have:

vαβ = ∂α∂βv −N Γkαβ∂kv, ∀α, β = 1, 2,(22)

(wherevαβ are the second order covariant derivatives ofv). Inserting (22) in the lastintegral on the righthand side of the equation (21) we obtain:

∫U

〈∇f,∇(v ◦ ϕ)〉dµg =∫U

〈∇(f.[(∂αv) ◦ ϕ]),∇ϕα〉dµg−

∫U

f.(vαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg−

∫U

f.[(∂kv) ◦ ϕ](NΓkαβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg.

(23)

ϕ is supposed harmonic so it is weakly harmonic. Consequently, the first and the thirdintegral in the righthand side of (23) are equal, so:

∫

U

〈∇f,∇(v ◦ ϕ)〉dµg = −

∫

U

f.(vαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg.

15

Take nowv = ψi. Then fori = 1, 2 we have:∫

U

〈∇f,∇(ψi ◦ ϕ)〉dµg = −

∫

U

f.(ψiαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg.

We compute the righthand side of the last equality:∫U

f.(ψiαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg =

=∫U

f.n∑

A,B=1

( ∂2ψi

∂xA∂xB◦ ϕ)〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψi

∂xA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψi

∂yA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψi

∂yA∂yB◦ ϕ)〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉dµg.

For example, fori = 1 we have:∫U

f.(ψ1αβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg =

∫U

f.n∑

A,B=1

( ∂2ψ1

∂xA∂xB◦ ϕ)〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψ1

∂xA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψ1

∂yA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψ1

∂yA∂yB◦ ϕ)〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉dµg.

Inserting Cauchy-Riemann equations associated toψ1 andψ2 (becauseψ is holomorphic)in the second and third sums of the righthand side of the aboveequality, we obtain:

∫U

f.(ψ1αβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg =

∫U

f.n∑

A,B=1

( ∂2ψ1

∂xA∂xB◦ ϕ)〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

(− ∂2ψ2

∂xA∂xB◦ ϕ)〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψ2

∂yA∂yB◦ ϕ)〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

∫U

f.n∑

A,B=1

( ∂2ψ1

∂yA∂yB◦ ϕ)〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉dµg

16

By easy computations we obtain:

∫U

f.(ψ1αβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg =∫U

f.n∑

A,B=1

[( ∂2ψ1

∂xA∂xB◦ ϕ)− ( ∂2ψ1

∂yA∂yB◦ ϕ)]

[〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉 − 〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉]dµg−∫U

f.n∑

A,B=1

[( ∂2ψ2

∂xA∂xB◦ ϕ)− ( ∂2ψ2

∂yA∂yB◦ ϕ)]

[〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉+ 〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉]dµg+∫U

f.n∑

A,B=1

[( ∂2ψ1

∂xA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉+

( ∂2ψ1

∂yA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉]dµg+

∫U

f.n∑

A,B=1

[( ∂2ψ2

∂xA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉−

( ∂2ψ2

∂yA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉]dµg.

Because the mapϕ is supposed pseudo-horizontally weakly conformal, the first and thesecond integral of the righthand side in the last equality are zero, so:

∫U

f.(ψ1αβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg =

∫U

f.n∑

A,B=1

[( ∂2ψ1

∂xA∂xB◦ ϕ) + ( ∂2ψ1

∂yA∂yB◦ ϕ)]〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉dµg+

+∫U

f.n∑

A,B=1

[( ∂2ψ2

∂xA∂xB◦ ϕ) + ( ∂2ψ2

∂yA∂yB◦ ϕ)]〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉dµg

Now, by Remark (4.4) the last two sums are zero. So we obtain:∫

U

〈∇f,∇(ψ1 ◦ ϕ)〉dµg = 0.

By a similar computation we can also prove:∫

U

〈∇f,∇(ψ2 ◦ ϕ)〉dµg = 0.

Thusψ ◦ ϕ is weakly harmonic and so, harmonic (cf. subsection 2.3.1).” ⇐ ”:Conversely, supposeψ ◦ ϕ : X → R

2 is harmonic for anyψ ∈ Holom(N). It isknown that the harmonicity ofψ ◦ϕ is equivalent (cf. subsection 2.3.1 and Lemma 4.3) tothe weak harmonicity ofψ ◦ϕ. So, for a given chartη : R2 → R

2, and for any quasiopensetU ⊂ ϕ−1(R2) of compact closure inX, the weak harmonicity reads:

∫

U

〈∇f,∇(ψi ◦ ϕ)〉dµg = 0,(24)

for i = 1, 2 and every functionf ∈ W 1,2c (U), whereψ = ψ1 + iψ2:.

17

Now, choose local complex holomorphic coordinates inN , (z1, ..., zn), wherezk =xk + iyk, ∀k = 1, ..., n. Takeψ = zk, ∀k = 1, ..., n, and denote

x1, ..., xn, y1, ..., yn‖ ‖ ‖ ‖ξ1 ξn ξn+1 ξ2n

by a generic termξk, ∀k = 1, ..., 2n. In particular, the equation (24), for the coordinatesfunctionsξk, reads:

∫

U

〈∇f,∇(ξk ◦ ϕ)〉dµg = 0, ∀k = 1, ..., 2n.(25)

For the domain chartV of (zk), any quasiopen setU ⊂ ϕ−1(V ) of compact closure inXand forf ∈ W 1,2

c (U) we have:

0 =∫U

〈∇f,∇(ξk ◦ ϕ)〉dµg =∫U

〈∇(f.[(∂αξk) ◦ ϕ]),∇ϕα〉dµg−

∫U

f.(ξkαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg−

∫U

f.[(∂γξk) ◦ ϕ](NΓγαβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg,

∀α, β, γ = 1, ..., 2n.

Taking into account that:∀k = 1, ..., 2n and∀α, β = 1, ..., 2n,

ξkαβ = 0 and∂αξk =

{0, if α = k1, if α 6= k

,

we obtain:

0 =∫U

〈∇f,∇(ξk ◦ ϕ)〉dµg =

=∫U

〈∇f,∇ϕk〉dµg −∫U

f.(NΓkαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg,

So,∀α, β, k = 1, ..., 2n,∫

U

〈∇f,∇ϕk〉dµg =

∫

U

f.(NΓkαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg.

This means thatϕ is weakly harmonic. Butϕ is a continuous map of classW 1,2loc (X,N),

so it is a harmonic map (cf. subsection 2.3.1).Now, for anyv : N → R such thatv ∈ C2(V ), whereV is a domain chart onN , we

have:∫U

〈∇f,∇(v ◦ ϕ)〉dµg =∫U

〈∇(f [(∂kv) ◦ ϕ]),∇ϕk〉dµg−

∫U

f(vαβ ◦ ϕ)〈∇ϕα,∇ϕβ〉dµg−

∫U

f [(∂kv) ◦ ϕ](NΓkαβ ◦ ϕ)〈∇ϕ

α,∇ϕβ〉dµg,

(26)

18

for α, β = 1, ..., 2n, whereU ⊂ ϕ−1(V ) is quasiopen with compact closure inX.In particular, for any holomorphic chart:

η : V ⊂ N → Cn

p 7→ (z1, ..., zn), zj = xj + iyj,

and for any local holomorphic functionψ : N → C,ψ = ψ1+iψ2, denotingx1, ..., xn, y1, ..., ynby ξγ , ∀γ = 1, ..., 2n, we apply equation (26) toψ1 andψ2 respectively:

0 =∫U

〈∇(f [(∂kψi) ◦ ϕ]),∇(ξk ◦ ϕ)〉dµg−

∫U

f(ψiαβ ◦ ϕ)〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉dµg−∫U

f [(∂kψi) ◦ ϕ](NΓkαβ ◦ ϕ)〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉dµg, ∀i = 1, 2.

In the above equality, the first and the last integral are equal (becauseϕ is harmonic), sowe have:

0 =

∫

U

f(ψiαβ ◦ ϕ)〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉dµg.

In the proof of the ”if” part we have obtained:

0 =∫U

f(ψ1αβ ◦ ϕ)〈∇(ξα ◦ ϕ),∇(ξβ ◦ ϕ)〉dµg =

=∫U

f.n∑

A,B=1

[( ∂2ψ1

∂xA∂xB◦ ϕ)− ( ∂2ψ1

∂yA∂yB◦ ϕ)]

[〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉 − 〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉]dµg−∫U

f.n∑

A,B=1

[( ∂2ψ2

∂xA∂xB◦ ϕ)− ( ∂2ψ2

∂yA∂yB◦ ϕ)]

[〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉+ 〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉]dµg+∫U

f.n∑

A,B=1

[( ∂2ψ1

∂xA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(yB ◦ ϕ)〉+

( ∂2ψ1

∂yA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(xB ◦ ϕ)〉]dµg+

∫U

f.n∑

A,B=1

[( ∂2ψ2

∂xA∂xB◦ ϕ)〈∇(yA ◦ ϕ),∇(xB ◦ ϕ)〉−

( ∂2ψ2

∂yA∂yB◦ ϕ)〈∇(xA ◦ ϕ),∇(yB ◦ ϕ)〉]dµg.

(27)

Choosing particular holomorphic functions as:ψ = zAzB andψ = izAzB, ∀A,B =1, ..., n, in equation (27), we obtain the pseudo-horizontally weakly conformal conditions(see Definition 3.1). �

A straightforward result deduced from the Theorem 4.2 is thefolowing:

Corollary 4.5 Letϕ : X → N be a pseudo-horizontally weakly conformal map from aRiemannian polyhedron into a Kahler manifold. Thenϕ is harmonic if and only if thecomponentsϕk = zk ◦ ϕ in terms of any holomorphic coordinates(zk)k=1,...,n in N , areharmonic maps (ϕk is understood as a map intoR2).

Proof: The ”only if part” is already proved through the proof of the Theorem 4.2.The ”if part”: Suppose that the mapϕ is harmonic. So it is pseudo harmonic mor-

phism. By Theorem 4.2,ϕ pulls back local holomorphic functions to local harmonicmaps which applies to any holomorphic coordinates inN . �

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We turn our attention to the relation between pseudo harmonic morphism on Rieman-nian polyhedron and local holomorphic maps between Kahlermanifolds. More precisely,we have:

Theorem 4.6 Let (X, g) be a Riemannian polyhedron and(N, JN , gN), (P, JP , gP ) betwo Kahler manifolds withdimCN = n anddimCP = p. A continuous mapϕ : X → N ,of classW 1,2

loc (X,N), is pseudo harmonic morphism if and only ifψ ◦ ϕ : X → P is a(local) pseudo harmonic morphism for all local holomorphicmapsψ : N → P .

Proof: ” ⇒ ” :Suppose thatϕ : X → N is a pseudo harmonic morphism and letψ : X → P be any

local holomorphic map.By Proposition 3.6,ψ ◦ ϕ is a pseudo-horizontally weakly conformal map.Let (z1, ..., zp) be local holomorphic coordinates inP , and setψj = zj ◦ ψ. On one

hand, the mapϕ is supposed PHM and obviously the complex functionsψj : P → C,∀j = 1, ..., p, are holomorphic so, by Theorem 4.2,ψj ◦ ϕ are harmonic,∀j = 1, ..., p.

On the other hand,ψ ◦ϕ is pseudo-horizontally weakly conformal. Thus by Corollary4.5, the mapψ ◦ ϕ is (local) harmonic.

” ⇐ ” :Suppose now thatψ ◦ ϕ : X → P is a (local) pseudo harmonic morphism, for any

local holomorphic mapψ : N → P . By Proposition 3.6, the mapϕ is pseudo-horizontallyweakly conformal.

As we have already done in the proof of Proposition 3.6, everylocal holomorphicfunction v : N → C can be obtained as a coordinate of some local holomorphic mapψv : N → P .

ψv ◦ ϕ is local a pseudo harmonic morphism. By Corollary 4.5,v ◦ ϕ is harmonic(because it is a coordinate ofψv ◦ ϕ). Therefore, we have shown that for any local holo-morphic functionv : N → C, the mapv◦ϕ is harmonic. So by Theorem 4.2 we concludethatϕ is a pseudo harmonic morphism. �

The following proposition will play an important roll in thenext section for construct-ing examples of PHM on admissible Riemannian polyhedra.

Proposition 4.7 Let (X, g) and (Y, h) be two admissible Riemannian polyhedra, and(N, JN , gN) a Kahler manifold without boundary of complex dimensionn. Letϕ : Y →N be a continuous map of classW 1,2

loc (Y,N), π : X → Y a proper, surjective, continuousmap of classW 1,2

loc (X, Y ) andϕ = ϕ ◦ π. If π is a harmonic morphism, thenϕ is pseudoharmonic morphism if and only ifϕ is pseudo harmonic morphism.

Proof: Suppose thatϕ : X → N is a pseudo harmonic morphism; by Theorem 4.2,it is equivalent to the fact thatϕ pulls back local holomorphic functions onN to localharmonic functions onX. So, for anyψ ∈ Holom(N), ψ = ψ1 + iψ2, the mapψ ◦ ϕ ∈W 1,2loc (X,R

2) is locally harmonic. But this can also reads: for anyψ ∈ Holom(N),ψ = ψ1 + iψ2, the functionsψ1 ◦ ϕ andψ2 ◦ ϕ are locally harmonic (becauseψ ◦ ϕ iscontinuous and the Christoffel symbols relative to the fixedchart ofR2 are all zero [cf.subsection 2.3.1]); or, also for anyψ ∈ Holom(N), with ψ = ψ1 + iψ2, the functions(ψ1 ◦ ϕ) ◦ π and(ψ2 ◦ ϕ) ◦ π are locally harmonic.

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On the other hand, the mapπ is a harmonic morphism and it is supposed surjectiveand proper so, by subsection 2.3.2 (cf. [7], Theorem 13.1), for every open setV ⊂ Y andany functionv : V → R, the functionv is harmonic if and only ifv ◦ π is harmonic. Inparticular this fact applies to the functions(ψ1 ◦ ϕ) and(ψ2 ◦ ϕ); in other words, for anylocal holomorphic functionψ = ψ1 + iψ2 onN , the functions(ψ1 ◦ ϕ) ◦ π respectively(ψ2 ◦ϕ)◦π are locally harmonic if and only if the functions(ψ1 ◦ϕ) respectively(ψ2 ◦ϕ)are locally harmonic.

The last assertion is equivalent to the fact that, for anyψ ∈ Holom(N), the mapψ ◦ϕis locally harmonic. But by Theorem 4.2, this means (iff) that ϕ : Y → N is a pseudoharmonic morphism. This ends the proof of the proposition.

�

5 Some examples.

In this short section we will offer some examples of pseudo harmonic morphisms onRiemannian polyhedra.

As we have shown in Proposition 3.4, every horizontally weakly conformal map froma Riemannian admissible polyhedron into a Hermitian manifold, is pseudo horizontallyweakly conformal. So every harmonic morphism into a Kahlermanifold is pseudo har-monic morphism. For other nontrivial examples, when the source polyhedra are smoothRiemannian manifolds, see for example [1].

Thanks to Proposition 4.7, we can derive a several non-obvious examples of pseudoharmonic morphisms on (singular) Riemannian polyhedra.

It is known (see Example 8.12, [7]) that given aK compact group of isometries of acomplete smooth Riemannian manifoldM , andπ : M → M/K the projection onto theorbit spaceM/K (with the quotient topology), there is a smooth triangulation ofM forwhichπ induces an admissible Riemannian polyhedral structure onM/K. The associatedintrinsic distancedM/K(y1, y2) between elementsy1, y2 of M/K, equals the intrinsicdistance inM between the corresponding compact orbitsπ−1(y1) andπ−1(y2), and thepolyhedronM/K with the distancedM/K is a geodesic space. The polyhedral structuredetermines a Brelot harmonic sheafHM/K onM/K (cf. Theorem 7.1, [7]). Moreover,πcaries the Brelot harmonic sheafHM onto a Brelot harmonic sheafH′

M/K = π∗HM onM/K andπ : (M,HM) → (M/K,H′

M/K) becomes a harmonic morphism, surjectiveand proper.

This construction can be applied to Riemannian orbifolds (cf. Subexample 8.13(ii),[7]) as follows:

LetM be a Riemannian manifold, andSr the symmetry group onr factors.Denote:

SrM := (M ×M × ...×M︸ ︷︷ ︸)/Sr

r -times

ther-fold symmetric power of the manifoldM .The compact groupSr acts isometrically, soSrM becomes a Riemannian orbifold

(singular if the dimension ofM > 3). Thus, as above, we obtain a proper, surjective,harmonic morphism fromM ×M × ...×M to SrM.

21

In the particular case whenM = Ck+s, with 2(k + s) > 3, following [1], one can

construct a pseudo harmonic morphism:

η : Ck ×Cs → C

r given by

(u, v) 7→(F1(u)P1(v)G1(u)Q1(v)

, ..., Fr(u)Pr(v)Gr(u)Qr(v)

),

whereF1,...,Fr,G1,...,Gr are homogenous polynomials onCk andP1,...,Pr,Q1,...,Qr arehomogenous polynomials onCs, all having the same degree. We know that the sum oftwo PHM is also a PHM, so we define:

ϕ : Ck+s ×Ck+s → C

r, ϕ(u, v) := η(u) + η(v),

for anyu, v ∈ Ck+s.

Using the harmonic morphismCk+s ×Ck+s → S2Ck+s, the mapϕ factors through a

PHM fromS2Ck+s to Cr.

Moreover, using the harmonic morphismCk+s → CP k+s−1, the mapη factors (see[1]) through a PHM fromCP k+s−1 to C

r which is neither holomorphic nor antiholo-morphic. Now using this map and apply the same arguments as before, we get a pseudoharmonic morphism fromS2CPk+s−1 to C

r.

Acknowledgments.We thank the ICTP Trieste for hospitality during this work.

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