Igv2008

Harmonic trinoids in complex projective spaces

Shimpei Kobayashi, Hirosaki University

12/12, 2008

IntroductionHarmonic maps into complex projective spaces

PreliminariesHarmonic spheresHarmonic tori

Equivariant harmonic maps in CPn

Isomorphisms between loop algebrasPotentials for equivariant harmonic maps

Harmonic trinoids in CPn

DPW methodSystem of ODEs and a scalar ODEHypergeometric equationsUnitarizability and interlace on the unit circleOpen problems

Let (M, g) and (N, h) be Riemannian manifolds and

Ψ : (M, g) → (N, h)

a C∞ map.Define

E(Ψ) =

∫

M|dΨ|2dVg,

where the norm is defined by g and h, and dVg is the volume formof M.

Let (M, g) and (N, h) be Riemannian manifolds and

Ψ : (M, g) → (N, h)

a C∞ map.Define

E(Ψ) =

∫

M|dΨ|2dVg,

where the norm is defined by g and h, and dVg is the volume formof M.Consider the variation Ψt for Ψ.

Ψ is harmonicdef⇔

d

dtE(Ψt)|t=0 = 0 ⇔ τ (Ψ) = 0,

where τ (Ψ) = trace∇dΨ is the tension field.

In particular, if dim M = 2, then the harmonicity can be writtenas

∇Ψ∂∂ z

dΨ(∂

∂z) = 0, (1)

where z = x + iy and (x, y) is a conformal coordinate.

Harmonic spheres

If M = S2, the followings (N, h) were studied in details:

I Sn (RPn) (Calabi, Chern)

I CPn (D. Burns, Eells-Wood, Din-Zakrzewski, Glaser-Stora)

I Gr2(Cn) (Chern-Wolfson, Burstall-Wood)

I Grk(Cn) (Wolfson, Wood)

These are based on

1) Holomorphic differential on S2 is zero

2) Techniques of Hermitian vector bundles.

Harmonic tori

If M = T2, the followings (N, h) were studied in details :

I S2 (Pinkall-Sterling)

I S3 (Hitchin)

I S4 (Pinkall-Ferus-Sterling-Pedit)

I Sn, CPn (Burstall, McIntosh)

I Gr2(C4), HP3 (Udagawa)

I Rank 1 compact symmetric spaces(Burstall-Ferus-Pedit-Pinkall)

These are based on integrable system methods.

Goal of this talk

I would like to discuss harmonic maps from M = S1 × R orM = CP1 \ 0, 1, ∞ into N = CPn.

Goal of this talk

I would like to discuss harmonic maps from M = S1 × R orM = CP1 \ 0, 1, ∞ into N = CPn.

Consider a C∞ map Ψ from a Riemann surface M into asymmetric space G/K:

∇Ψ∂∂ z

dΨ(∂

∂z) = 0 ⇔

dαk + 12[αk ∧ αk] = −[α′

p ∧ α′′p ] = 0,

dα′p + [αk ∧ α′

p] = 0,

⇔dαλ + 1

2[αλ ∧ αλ] = 0,

αλ = λ−1α′p + αk + λα′′

p , λ ∈ S1.

where α = F−1dF is the Maurer-Cartan form of a liftF : M → G, g = k ⊕ p and TMC = T′M + T′′M.

Equivariant harmonic maps in k-symmetric spaces

DefinitionA map Ψ : R2 → G/K is called R-equivariant if

Ψ(x, y) = exp(xA0)Φ(y),

for some A0 ∈ g and Φ : R → G/K.

Theorem (Burstall-Kilian)

All equivariant primitive harmonic maps in k-symmetric spacesG/K (with an order k-automorphism τ ) are constructed fromdegree one potentials:

ξ = λ−1ξ−1 + ξ0 + λξ1 ∈ Λgτ , (2)

where Λgτ = ξ : S1 → g | ξ(e2πi/kλ) = τξ(λ) is the loopalgebra of the Lie algebra g of G, ξj ∈ gj and ξj = ξ−j with theeigenspace decomposition of gC =

∑

i∈Zkgi.

Equivariant harmonic maps in CPn

For CPn case, G = SU(n + 1) with the involutionσ = Ad diag [1, −1, . . . , −1], thus K = S(U(1) × U(n)) andGC = SL(n + 1, C).

It is known that harmonic maps in CPn can be classified into

I isotropic,

I non-isotropic weakly conformal with isotropic dimensionr ∈ 1, . . . , n − 1,

I non-conformal.

Problem: Which degree one potentials are corresponding to theabove cases?

Isomorphism

Lemma (Pacheco)

Let g be a Lie algebra, τ : g → g an automorphism of order kand σ : g → g an involution.Define Γ as a map between Λgτ and Λgσ

Γ(ξ)(λ) = s(λ)t(λ−2/k)ξ(λ2/k) ∈ Λgσ for ξ ∈ Λgτ , (3)

where t : S1 → Aut g and s : S1 → Aut g are automorphismsuch that t(e2πi/k) = τ and s(−1) = σ respectively.Then Γ is an isomorphism.

Isomorphism

Lemma (Pacheco)

Let g be a Lie algebra, τ : g → g an automorphism of order kand σ : g → g an involution.Define Γ as a map between Λgτ and Λgσ

Γ(ξ)(λ) = s(λ)t(λ−2/k)ξ(λ2/k) ∈ Λgσ for ξ ∈ Λgτ , (3)

where t : S1 → Aut g and s : S1 → Aut g are automorphismsuch that t(e2πi/k) = τ and s(−1) = σ respectively.Then Γ is an isomorphism.

Let t and s be t(λ) = Addiag[1, λ, . . . , λk−2, λk−1, . . . , λk−1]and s(λ) = Ad diag[1, λ, . . . , λ] respectively. Then it is easy tosee t(e2πi/k) = τ and s(−1) = σ. Define Γ as in (3), and let

ξ = λ−1ξ−1 + ξ0 + λξ−1t∈ Λsu(n + 1)τ

be the degree one potential.

Proposition

A harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential

η = Γ(ξ) ∈ Λsu(n + 1)σ, (4)

where the order k of τ and the degree one potential ξ are given asfollows:

Proposition

A harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential

η = Γ(ξ) ∈ Λsu(n + 1)σ, (4)

where the order k of τ and the degree one potential ξ are given asfollows:

(a) if it is isotropic:

k = n + 1 and ξ−1 is principal nilpotent.

Proposition

A harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential

η = Γ(ξ) ∈ Λsu(n + 1)σ, (4)

where the order k of τ and the degree one potential ξ are given asfollows:

(a) if it is isotropic:

k = n + 1 and ξ−1 is principal nilpotent.

(b) if it is non-isotropic weakly conformal with the isotropicdimension r ∈ 1, 2, · · · , n − 1:

k = r + 2 ∈ 3, 4, · · · , n + 1 and ξ−1 is semisimple.

Proposition

A harmonic map in CPn is R-equivariant if and only if it isgenerated by the following degree one potential

η = Γ(ξ) ∈ Λsu(n + 1)σ, (4)

where the order k of τ and the degree one potential ξ are given asfollows:

(a) if it is isotropic:

k = n + 1 and ξ−1 is principal nilpotent.

(b) if it is non-isotropic weakly conformal with the isotropicdimension r ∈ 1, 2, · · · , n − 1:

k = r + 2 ∈ 3, 4, · · · , n + 1 and ξ−1 is semisimple.

(c) if it is non-conformal:

k = 2 and ξ−1 is semisimple.

Equivariant harmonic maps in CP1

Figure: These figures are created by Nick Schmitt.

Loop groups

Definition

G : A compact simple Lie group, g : Lie algebra of G,

GC : The complexification of G, gC : Lie algebra of GC,

σ : A involution of G, K : The fixed point set of σ

k : Lie algebra of K, g = k ⊕ p : Direct sum

B : The solvable part of an Iwasawa decomposition

KC = K · B, K ∩ B = e

Loop groups

ΛGσ := H : S1 → G | σH(λ) = H(−λ),

Λgσ := h : S1 → g | σh(λ) = h(−λ)

ΛgCσ := h : S1 → gC | σh(λ) = h(−λ)

ΛGC

σ := H : S1 → GC | σH(λ) = H(−λ)

Λ+BGC

σ :=

H+ ∈ ΛGC

σ |H+ can be extend holomorphically

to D1 and H+(0) ∈ B

Loop groups

ΛGσ := H : S1 → G | σH(λ) = H(−λ),

Λgσ := h : S1 → g | σh(λ) = h(−λ)

ΛgCσ := h : S1 → gC | σh(λ) = h(−λ)

ΛGC

σ := H : S1 → GC | σH(λ) = H(−λ)

Λ+BGC

σ :=

H+ ∈ ΛGC

σ |H+ can be extend holomorphically

to D1 and H+(0) ∈ B

We assume that the coefficients of all g ∈ Λgσ are in the Wieneralgebra

A =

f(λ) =∑

n∈Z

fnλn : Cr → C ;

∑

n∈Z

|fn| < ∞

.

The Wiener algebra is a Banach algebra relative to the norm‖f‖ =

∑

|fn|, and A consists of continuous functions.

DPW method

Step1 η(z, λ) =∑

∞

k=−1 λkξk(z) : ΛgC

σ-valued 1-form on Σ ⊂ C,

where ξeven(z) ∈ Ω1,0(kC) and ξodd(z) ∈ Ω1,0(pC).

Step2 Solve ODE dC = Cη.

Step3 Iwasawa decomposition : C = FW+, F : Σ → ΛGσ andW+ : Σ → Λ+

BGC

σ.

Step4 Projection π F|λ∈S1 : Σ → G/K, where π : G → G/K.

Theorem (Dorfmeister-Pedit-Wu)

Multiplication ΛGσ × Λ+BGC

σ → ΛGC

σ is a diffeomorphism onto.

Theorem (Dorfmeister-Pedit-Wu, 1998)

Every harmonic map from a simply connected domain Σ into G/Kcan be constructed in this way.

DPW method

Step1 η(z, λ) =∑

∞

k=−1 λkξk(z) : ΛgC

σ-valued 1-form on Σ ⊂ C,

where ξeven(z) ∈ Ω1,0(kC) and ξodd(z) ∈ Ω1,0(pC).

Step2 Solve ODE dC = Cη.

Step3 Iwasawa decomposition : C = FW+, F : Σ → ΛGσ andW+ : Σ → Λ+

BGC

σ.

Step4 Projection π F|λ∈S1 : Σ → G/K, where π : G → G/K.

Theorem (Dorfmeister-Pedit-Wu)

Multiplication ΛGσ × Λ+BGC

σ → ΛGC

σ is a diffeomorphism onto.

Theorem (Dorfmeister-Pedit-Wu, 1998)

Every harmonic map from a simply connected domain Σ into G/Kcan be constructed in this way.

From now on, CPn is represented as the symmetric spaceU(n + 1)/U(1) × U(n) with the involutionσ = Ad diag [1, −1, . . . , −1].

System of ODEs and a scalar ODEConsider

ν, τj ∈

∞∑

k=1

λ2k−3g2k−3

∣

∣

∣

∣

∣

g2k−3 is a holomorphic function on M

,

where i = 1, . . . , n and(

dn+1

dzn+1−

ν′

ν

dn

dzn− ντ1

dn−1

dzn−1− · · · − ντn

)

u = 0. (5)

System of ODEs and a scalar ODEConsider

ν, τj ∈

∞∑

k=1

λ2k−3g2k−3

∣

∣

∣

∣

∣

g2k−3 is a holomorphic function on M

,

where i = 1, . . . , n and(

dn+1

dzn+1−

ν′

ν

dn

dzn− ντ1

dn−1

dzn−1− · · · − ντn

)

u = 0. (5)

Setu1, . . . un+1 : A fundamental solutions of (5),

C :=

u(n)1ν

u(n−1)1 · · · u

(0)1

......

. . ....

u(n)n+1

νu

(n−1)n+1 · · · u

(0)n+1

,

where u(0)j = uj and u

(k)j = dku

dzk , (k > 0).

Lemma

(1)

η := C−1dC =

0 ν 0 · · · 0

τ1 0 1 · · · 0...

.... . .

. . ....

......

. . .. . . 1

τn 0 · · · · · · 0

. (6)

(2) η =

∞∑

k=−1

λkξk is a holomorphic potential on M, where

ξeven ∈ Ω1,0(kC) and ξodd ∈ Ω1,0(pC).

Fact: Monodromy representations of (5) and (6) are the same.

Hypergeometric functions

n+1Fn(α1, . . . , αn+1; β1, . . . , βn|z)

=

∞∑

k=0

(α1)k · · · (αn+1)k

(β1)k · · · (βn)kk!zk, (7)

where α1, . . . , αn+1, β1, . . . , βn ∈ C, (x)k is the Pochhammersymbol or rising factorial

(x)k =Γ(x + k)

Γ(x)= x(x + 1) · · · (x + k − 1).

I n+1Fn(α1, . . . , αn+1; β1, . . . , βn|z) is called thehypergeometric function n+1Fn.

I 2F1(α1, α2; β1|z) is the Gauß’s hypergeometric function.

Let D(α; β) = D(α1 . . . αn+1; β1 . . . βn+1) be the differentialoperator

D(α; β) = (θ+β1−1) . . . (θ+βn+1−1)−z(θ+α1) . . . (θ+αn+1)

for α1, . . . , αn+1, β1, . . . , βn+1 ∈ C, where θ = z ddz

. Thehypergeometric equation is defined by

D(α; β)u = 0.

Let D(α; β) = D(α1 . . . αn+1; β1 . . . βn+1) be the differentialoperator

D(α; β) = (θ+β1−1) . . . (θ+βn+1−1)−z(θ+α1) . . . (θ+αn+1)

for α1, . . . , αn+1, β1, . . . , βn+1 ∈ C, where θ = z ddz

. Thehypergeometric equation is defined by

D(α; β)u = 0.

Local exponents around the points z = 0, ∞, 1 are

z = 0 z = ∞ z = 11 − β1 α1 01 − β2 α2 11 − β3 α3 2

......

...

1 − βn+1 αn+1 γ =

n+1∑

1

βj −n+1∑

1

αj − 1

Fact: D(α; β)u = 0 is well-defined on CP1 \ 0, 1, ∞.

If βi are distinct mod Z, n + 1 independent solutions ofD(α; β)u = 0 are given by

z1−βin+1Fn(1+α1−βi, . . . , 1+αn+1−βi; 1+β1−βi,

∨. . ., 1+βn+1−βi|z),

where i = 1, . . . , n + 1 and ∨ denotes omission of 1 + βi − βi.

I V(α; β):The local solution space of D(α; β)u = 0 around z0.

I G : The fundamental group π1(CP1 \ 0, 1, ∞, z0).

I M(α, β) : G → GL(V(α; β)) : Monodromy representationof D(α; β)u = 0.

Theorem (Beukers-Heckman, 1989)

Let M(α; β) be the Monodromy group of D(α; β)u = 0. Then

M(α; β) are simultaneously conjugated into U(n + 1).m iff

0 < α1 < β1 < α2 < β2 < . . . < αn+1 < βn+1 5 1or0 < β1 < α1 < β2 < α2 < . . . < βn+1 < αn+1 5 1 .

Remarkαj and βj are determined by solving the indicial equations, whichare n-th order algebraic equations.

There are several problems for an application to harmonic maps inCPn.

I αj and βj depend on the additional parameter λ ∈ C.

I αj and βj need to be real and satisfy the inequality for almostall λ ∈ S1.

I Products and sums of αj and βj are ν and τj as in theholomorphic potential of (6).

The case n = 1 (Gauß’s hypergeometric equation)

Local exponents

z = 0 z = ∞ z = 11 − β1 α1 0

1 − β2 α2 γ =2∑

1

βj −2∑

1

αj − 1

Set

α1 = 1 − v1 − v2 − v3, α2 = 1 − v1 − v2 + v3,

andβ1 = 1 − 2v1, β2 = 1,

where

vj =1

2−

1

2

√

1 + wj(λ − λ−1)2

Spherical triangle inequality

0 < α1 < β1 < α2 < β2 5 1 ⇔

v1 + v2 + v3 < 1v1 < v2 + v3

v2 < v1 + v3

v3 < v1 + v2

(8)

It is not difficult to show that the above inequality are satisfied forsome choices of wj. Moreover all problems can be solved(Kilian-Kobayashi-Rossman-Schmitt, Dorfmeister-Wu).

Remark

I Umehara-Yamada considered the similar inequality for CMCH=1 in H3. (No λ dependence!)

Examples of CMC trinoids in space forms

Figure: These figures are created by Nick Schmitt.

The case n > 1

Example

I For the isotropic case, αj and βj do not depend on λ. Thusthere exist isotropic harmonic trinoids in CPn.

I For n = 2, 3, the indicial equation can be solved explicitly.We can show that there exist examples of harmonic trinoids inCP2 and CP3.

Open problem

I What are behaviors around the punctures? Are theyasymptotically converge to equivariant ones?

I Prove the existence of non-isotropic harmonic trinoids forn = 4.