New way of constructing mapping class group invariant K ahler - - PowerPoint PPT Presentation

Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

New way of constructing mapping class group invariant K¨ ahler metrics on Teichm¨ uller space and energy of harmonic maps

CIRM-Teichm¨ uller Theory conference

Inkang Kim

(Joint with Wan and Zhang)

Korea Institute for Advanced Study School of Mathematics

2020-10-06

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Outline

1

Teichm¨ uller space

2

Energy of harmonic maps

3

Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace

4

Convexity of energy function for Σ → S

5

Harmonic maps for branched coverings and general remarks

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Σ closed surface of genus ≥ 2. X Riemann surface (either with complex structure or hyperbolic metric) Teichm¨ uller space T of Σ is the set of pairs f : Σ → X where f is homeo up to equiv relation (f, X) ∼ (g, Y) ⇔ ∃ biholo (or isometry)h : X → Y st h◦f ∼ g. =set of hyperbolic structures (complex structures, conformal structures) up to isotopy

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Teich distance dT((f, X), (g, Y)) = 1

2 log minh:X→Y K(h)

where h is quasi-conformal making the diagram commute up to htpy and K(h) is the quasi-conformal constant. Σ

f

• g
• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

X

h

⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦

Y Teich distance dT is Finslerian (not Riemannian)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Mapping class group, MCG(Σ) = {Orientation preserving self − homeos}/homeos isotopic to identity, acts on T via [φ](f, X) = (f ◦ φ, X) by changing the marking. MCG acts as isometries on (T , dT). (T , dT) not Gromov hyperbolic (Masur-Wolf) T is a complex mfd of complex dim=3g(Σ) − 3 where g(Σ) = genus of Σ. Furthermore it is Stein i.e. ∃ proper plurisubharmonic map

• n T . Indeed it is realized as pseudoconvex bounded

domain (L. Bers) but not convex (Markovic). (A domain G is pseudoconvex if G has a continuous plurisubharmonic exhaustion function.) The Weil-Petersson metric on T is Gromov-hyperbolic if and only the dimension of T is less than or equal to 2.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

X

π

• Xz = π−1(z) Riem. surf. w/ cx str z ∈ T loc holo coord v

T Teichm¨ uller space of Σ, local holo coord z = (z1, · · · , z3g−3) (z, v) local holomorphic coordinates of X around π−1(z) TX = V ⊕ H V =< ∂ ∂v >, H =< δ δzα = ∂ ∂zα +av

α

∂ ∂v >, V∗ =< δv = dv−av

αdzα >

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Xz equipped with K¨ ahler metric of K¨ ahler form ωXz = √ −1φv¯

vdv ∧ d¯

v = √ −1φv¯

v(dv ⊗ d¯

v − d¯ v ⊗ dv) eφ = φv¯

v = ∂v∂¯ vφ K¨

ahler Einstein equation hyperbolic metric Φz = φv¯

v(dv ⊗ d¯

v + d¯ v ⊗ dv)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

TzT

Kodaira−Spencer

H1(Xz, TXz) = H0(Xz, K 2

Xz)∗

Serre duality Harmonic Beltrami differentials

∂ ∂zα harm.hori.lift

• ∂

∂zα + av α ∂ ∂v →

∂v( ∂

∂zα + av α ∂ ∂v ) = Av α¯ v ∂ ∂v ⊗ d¯

v Kodaira-Spencer class

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Weil-Petersson metric

If the metric on Xz is locally λ(v)|dv| and φ, ψ ∈ T ∗

XzT quad diff,

then < φ, ψ >=

• Xz

φ¯ ψ λ2 dVXz

• r

ωWP = √ −1Gα¯

βdzα ∧ d¯

zβ Gα¯

β = ∂

∂zα , ∂ ∂zβ =

• Xz

(Av

α¯ v

∂ ∂v ⊗ d¯ v, Av

β¯ v

∂ ∂v ⊗ d¯ v)dVXz =

• Xz

Av

α¯ vAv β¯ v

√ −1φv¯

vdv ∧ d¯

v =

• Xz

c(φ)α¯

βdVXz.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Here √ −1∂¯ ∂φ = √ −1c(φ)α¯

βdzα ∧ d¯

zβ + √ −1φv¯

vδv ∧ δ¯

v c(φ)α¯

β = φα¯ β − φv¯ vφα¯ vφv¯ β > 0.

av

α = −φv¯ vφα¯ v, φv¯ v = φ−1 v¯ v

Av

α¯ v = ∂¯ vav α

Aα¯

v¯ v = Av α¯ vφv¯ v = −∇¯ vφα¯ v = −φα¯ v;¯ v

Aα¯

v¯ v;v = 0

(−φv¯

v∂v∂¯ v + 1)c(φ)α¯ β = Av α¯ vA¯ v ¯ βv.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Properties of Weil-Petersson metric

incomplete (Wolpert and Chu) K¨ ahler metric (Alfors), −∞ < negative unbounded sectional curvature < 0 (Tromba, Wolpert) Ricci curvature ≤ −

1 2π(g−1). (Tromba, Wolpert)

W-P volume of moduli space T /Mod(Sg) is calculated by Mirzakhani. Question: Are there better K¨ ahler metrics?

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

K¨ ahler hyperbolic metric

K¨ ahler hyperbolic (Gromov): complete K¨ ahler with bounded curvature and bounded K¨ ahler primitive, finite volume for Moduli space McMullen (Is his metric negatively curved?), Liu-Sun-Yau (Ricci metric and perturbed Ricci metric) created such K¨ ahler hyperbolic metrics K¨ ahler-Einstein metric (Cheng-Yau) is equivalent to Teichm¨ uller metric (Liu-Sun-Yau) and equiv to McMullen metric equiv to Ricci and perturbed Ricci metric.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Several ways to create W-P metric

Wolpert d2 dt2 log ℓ(Xt, g0)|t0 = 4 3 || ˙ X0||2

WP

area(X0). length on Xt of a random geodesic on X0 McMullen || ˙ φX||2

P = 4

3 || ˙ X0||2

WP

area(X0). Pressure metric pulls back to multiple of W-P under thermodynamic embedding

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Takhtajan-Teo-Zograf, Krasnov-Schlenker (DX(Re θ(c−, ·)))(Y) = X, YWP, θ(c−, ·) = 4∂ωM where ωM renormalized volume of quasi-fuchsian 3-mfd, c− Kahler potential for W-P on T∂+M. Fischer-Tromba, Wolf harmonic (homotopic to id) : Σ → Σ, √ −1∂¯ ∂ log(Energy) = const.ωWP.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

First and Second variation of Energy function

u : (M, g = (gij), {xi}) → (Σ, h, {v}) ∇ : ∧lT ∗M ⊗ u∗TΣ → ∧l+1T ∗M ⊗ u∗TΣ L2-inner product on ∧lT ∗M ⊗ u∗TΣ , =

• M

(, )dvolg ∇, = , ∇∗ △ = ∇∗∇ + ∇∇∗ Laplace operator=self adj, semi-pos elliptic

• perator

Harmonic form △ω = 0 ⇔ ker∇ ∩ ker∇∗

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Energy E(u) = 1

2||du||2 = 1 2

• M |du|2dvolg

A = Aα¯

v¯ v ¯

uiφv¯

vdxi ⊗ ∂ ∂v , first variation

∂E(u) ∂zα =

• M

Aα¯

v¯ v ¯

ui ¯ ujgijdvolg = A, du Second variation ∂2E(u) ∂zα∂¯ zα =

• M

(c(φ)α¯

αgijφv¯ vui ¯

uj + gijAα¯

v¯ v ¯

uj∇¯

zα¯

ui)dvolg

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

New way to construct K¨ ahler metrics

We adopt harmonic map approach to create K¨ ahler metrics on T . Theorem (Mn, g) Riemannian manifold, u0 : M → Σ fixed smooth map whose image is not a point. uz : M → Xz (unique) harmonic map homotopic to u0. E(z) = E(uz) energy of a harmonic map uz is a smooth map on T . Then ∂2E ∂zi∂¯ zj ξiξj > 0 i.e. E(z) is plurisubharmonic on T and furthermore log Ei(z) is also plurisubharnonic for different maps Mi → Σ. Hence the energy defines a K¨ ahler potential on T depending on M and u0. Also E(ut) is strictly convex along W-P geodesic in T .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Corollary If dim M = 1, then ∂2E1/2 ∂z∂¯ z = 1 2 1 E1/2

• M

( + 1)−1(|A|2)dµg + 1 2|du|2(|du|2 + ∆)−1A, A

• ,

where = −φv¯

v∂v∂¯ v and |A|2 = |Av z¯ v|2( 1 2|du|2) is a smooth

function on (z, v) = (z, u(z, x)). If we take the arc-length parametrization at z = z0, i.e. 1

2|du|2(z0) = 1, then the first and

the second variations of the geodesic length function are given by ∂ℓ(z) ∂z |z=z0 =1 2A, du ∂2ℓ(z) ∂z∂¯ z |z=z0 =1 2

• M

( + 1)−1(|A|2)dµg + (2 + ∆)−1A, A

• .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Idea of proof second variation= 1 2

• M

c(φ)α¯

α|du|2dvolg + [(△−1△ − ∇△−1∇∗)

+(∇△−1∇∗ − ∇(L − GL−1 ¯ G)−1∇∗ + H]A, A > 0 where H : A1(M, u∗TΣ) → ker△ harmonic projection, G = gijφv¯

vuiuj ∂ ∂v ⊗ d¯

v and L = △ + gijφv¯

vui ¯

• uj. Note

Al = Im△ ⊕ Ker△, Id = △−1△ + H.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Application: We can use convexity of energy function to reprove Nielsen realization theorem (Kerchkhoff), namely any finite subgroup of mapping class group of Σ can be realized as an isometry group of (Σ, h) for some hyperbolic metric h. When does this give W-P metric? Theorem uz : (M, ωg) → (Xz, Φz) harmonic map from K¨ ahler manifold (M, ωg). Suppose u0 = uz0 is (anti)-holomorphic. Then √ −1∂¯ ∂ log E(z)|z=z0 = ωWP 2π(g(Σ) − 1). Specially when M is a Riemann surface √ −1∂¯ ∂E(z)|z=z0 = 2|deg u(z0)|ωWP.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

One can use Σ also as a domain of a map u0 : Σ → M. But in this case the problem is a bit harder. One needs to assume M is Hermitian non-positively curved i.e. R(X, Y, ¯ X, ¯ Y) ≤ 0. Then we obtain a similar result. Theorem uz : (Xz, Φz) → M unique harmonic map hpic to u0. Then E(z) and log E(z) are plurisubharmonic on T . E(z) being plurisubharmonic is originally proved by D. Toledo in this case u : Σ → M. Somehow convexity of energy function in this case is much harder than u : M → Σ. We can prove (local) convexity of energy function at critical points only when M = S surface.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Convex real projective structures on surface

Higher Teichm¨ uller Theory P(S)=the space of strictly convex real projective structures

• n S.

= Hitchin component of the character variety χ(π1(S), SL(3, R)).(N. Hitchin, Higgs bundle) = Anosov representations (Labourie) = Positive representation (Fock-Goncharov)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

One can construct K¨ ahler metrics on P(S) as follows. P(S) is a holomorphic vector bundle over T (S) (Labourie, Loftin), H0(Xz, K 3

Xz)

• ❲

❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲

cubic holomorphic differentials P(S) = E Griffiths positive

• T (S)⊂ E∗ Griffith negative

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Theorem (K-Zhang) E∗ has a mapping class group invariant K¨ ahler metric which extends W-P metric on T (S). Indeed π∗ψ + || · ||2 gives K¨ ahler potential on E∗ where || · ||2 is L2 norm and ψ is K¨ ahler potential on T .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

In general, if π : E → M holomorphic vector bundle over K¨ ahler manifold (M, ω) with G Hermitian metric on E of Griffiths negative curvature i.e Ri¯

jα¯ βvi ¯

vjξα¯ ξβ < 0, then Ω = π∗ω + √ −1∂¯ ∂G is a K¨ ahler form on E where R is the Chern connection curvature, v fiber and ξ ∈ TM.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Embedding of Quasi-fuchsian space into the space of complex projective structure

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Properties of MCG(S)-inv K¨ ahler metrics

Curvature vanishes along vertical vectors R( ∂ ∂vk , ·)( ∂ ∂vi ), ∂ ∂vj = 0 P = vi ∂

∂vi tautological section

R(ξ, ¯ ξ)(P), P < 0 (T (S), WP), E = TT (S), Ricci curvature of Ω|T (S) ≤ − 1 π(g(S) − 1)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Harmonic maps between surfaces

Now we prove local convexity of energy function at critical points between surfaces Σ → S where the metric on S is fixed and the metric on Σ varies. Theorem If t0 ∈ T (Σ) is a critical point of energy function, then the energy function is convex at t0. If moreover, dut0 is never zero, then the energy function is strictly convex at t0. As an application, we have

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Corollary If u0 : Σ → S is a covering map, then there exists a unique complex structure t0 ∈ T such that the associated harmonic map ut0 is ± holomorphic, and E(t) ≥ E(t0) = Area(Σ). Moreover, the energy density satisfies 1

2|du|2(t0) ≡ 1. Indeed,

the unique hyperbolic metric on Σ which minimizes the energy is the pull-back hyperbolic metric via ut0. In this case, d2E(t) dt2 |t=0 = 4µ2

WP > 0,

where µ is a tangent vector (Beltrami differential) to a Weil-Petersson geodesic at t0.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Energy between surfaces

Let Σ be a Riemann surface of genus gΣ ≥ 2, and z = x + iy denote the local holomorphic coordinate on Σ. Denote by λ2(z)dzd¯ z := λ2 2 (dz ⊗ d¯ z + d¯ z ⊗ dz) = λ2(dx2 + dy2) the hyperbolic metric, i.e. its curvature satisfies K := − 4 λ2 ∂2 ∂z∂¯ z log λ = −1. The associated Hermitian metric is λ2dz ⊗ d¯ z, and the fundamental (1, 1)-form is given by ωΣ = i 2λ2dz ∧ d¯ z = λ2dx ∧ dy, (1) where dz ∧ d¯ z = dz ⊗ d¯ z − d¯ z ⊗ dz. The area of Σ is Area(Σ) :=

• ωΣ = −2πχ(Σ) = 4π(gΣ − 1).

(2)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Let S be also a Riemann surface of genus gS ≥ 2, and equipped with the hyperbolic metric ρ2(v)dvd¯ v = ρ2 2 (dv ⊗ d¯ v + d¯ v ⊗ dv), where v denotes the local holomorphic coordinate on S. Thus 4 ρ2 ∂2 ∂v∂¯ v log ρ = 1. (3) With respect to the local complex coordinates {v, ¯ v} on S, any smooth map u : Σ → S can be written as (v, ¯ v) = u(z) = (uv, u¯

v)

for some local smooth functions uv, u¯

• v. The energy of u is

E(u) :=

• Σ

ρ2(u(z))(|uv

z |2 + |uv ¯ z |2) i

2dz ∧ d¯ z, (4) where uv

z := ∂zuv and uv ¯ z := ∂¯ zuv.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

u is called harmonic if ∇¯

zuv z := ∂¯ zuv z + 2ρv

ρ uv

z uv ¯ z = 0,

(5) where ∇ denotes the natural induced connection on the complexified bundle u∗TCS =: u∗(TS ⊗ C) = u∗K ∗

S ⊕ u∗K ∗ S

from the Chern connection of the Hermitian line bundle (K ∗

S, ρ2dv ⊗ d¯

v).

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Energy function on Teichm¨ uller space

Let T = M−1/D0 be the Teichm¨ uller space of Σ, where M−1 is the space of all smooth Riemannian metrics on Σ with scalar curvature −1 and D0 is the group of all smooth orientation preserving diffeomorphisms of Σ in the identity homotopy class. For the fixed Riemann surface S and any hyperbolic metric g in M−1, since each smooth map homotopic to u0 (deg u0 0) is surjective, so there exists a unique harmonic map u(g) homotopy to u0;Thus the energy function E(g) := E(u(g)) is a smooth function on M−1. One can check that E(f ∗g) = E(g), for any f ∈ D0. Thus E descends to a smooth function on Teichm¨ uller space T := M−1/D0. We denote the energy function by E(t), t ∈ T .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Weil-Petersson geodesic

For any t ∈ T , denote by Σt = (Σ, t) the associated Riemann surface with the complex structure t ∈ T . Let ωΣt be the fundamental (1, 1)-form as in (1). The tangent space TtT is identified with the space of harmonic Beltrami differential µ = µ(z) d¯

z dz , and the L2-norm defines the Weil-Petersson metric

µ2

WP =

• Σ

|µ(z)|2ωΣt. (6) The Weil-Petersson metric is K¨ ahler, negatively curved, not

• complete. However, the synthetic geometry is quite similar to

that of a complete metric of negative curvature, and indeed Wolpert showed that every pair of points can be joined by a unique Weil-Petersson geodesic, and a geodesic length function is strictly convex along a Weil-Petersson geodesic.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Fix a point t0 ∈ T and let g0 = λ2

0dzd¯

z be the corresponding hyperbolic metric on Σt0. Let Γ(t), (Γ(0) = t0), be the Weil-Petersson geodesic arc with initial tangent vector given by the harmonic Beltrami differential µ = ¯

q λ2 d¯ z dz , where qdz2 is a

holomorphic quadratic differential on Σt0. Then the associated hyperbolic metrics on Σt has the following Taylor expansion near t = 0 (M. Wolf), g(t) =λ2

0dzd¯

z + t(qdz2 + qdz2) (7) + t2 2        2|q|2 λ4 − 2(∆ − 2)−1 2|q|2 λ4        λ2

0dzd¯

z + O(t4). (8) Here ∆ = 4

λ2 ∂2 ∂z∂¯ z = 1 λ2

0 (∂2

x + ∂2 y). Furthermore there is

a point-wise estimate for the term involving (∆ − 2)−1, α := −2(∆ − 2)−1 |q|2 λ4 ≥ 1 3 |q|2 λ4 . (9)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Energy function along Weil-Petersson geodesics

the Riemannian metric g(t) can be written as g(t) = (dz, d¯ z) ⊗ G dz d¯ z

• ,

where G is a matrix given by G = Gzz Gz¯

z

Gz¯

z

G¯

z¯ z

• =

            tq

λ2 2 + t2 2

• |q|2

λ4

0 + α

• λ2

λ2 2 + t2 2

• |q|2

λ4

0 + α

• λ2

tq             + O(t4). The volume element dµg(t) is given by dµg(t) := i

• | det G|dz ∧ d¯

z. (10)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Fix a smooth map u : Σ → S with deg u0 0. For each t, we get a harmonic map u = ut : (Σ, g(t)) → (S, ρ2dvd¯ v) which is homotopic to a fixed smooth map u0. The energy density is defined by 1 2|du|2 := 1 2Trg(t)(u∗(ρ2dvd¯ v)) = 1 2Tr(G−1U), (11) where du := uv

z dz ⊗ ∂

∂v + uv

¯ z d¯

z ⊗ ∂ ∂v + uv

¯ z dz ⊗ ∂

∂¯ v +uv

z d¯

z ⊗ ∂ ∂¯ v ∈ A1(Σ, u∗TCS) and the matrix U is U = ρ2       uv

z uv ¯ z 1 2(|uv z |2 + |uv ¯ z |2) 1 2(|uv z |2 + |uv ¯ z |2)

uv

¯ z uv z

      .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Then 1 2|du|2 = 1 2 ρ2(u(z)) det G

• Gzzuv

¯ z uv z + G¯ z¯ zuv z uv ¯ z − Gz¯ z(|uv z |2 + |uv ¯ z |2)

• .

(12) Hence, the energy function along the Weil-Petersson geodesic Γ(t) is E(t) = 1 2

• Σ

|du|2dµgt. (13)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

The first derivative

We obtain dE(t)

dt |t=0

= Re

• Σ

      −4ρ2 λ2 ¯ quv

¯ z uv z

       i 2dz∧d¯ z = −4

• ρ2uv

z uv ¯ z dz2, ¯

q λ2 d¯ z dz

• QB

, (14) where ·, ·QB is a pairing between holomorphic quadratic differentials and harmonic Beltrami differentials by φdz2, µd¯ z dz QB := Re

• Σ

φµ i 2dz ∧ d¯ z, which is non-degenerate. Note here that ρ2uv

z uv ¯ z dz2 is a

holomorphic quadratic differential (called Hopf differential of u) since ∂¯

z(ρ2uv z uv ¯ z ) = ρ2∇¯ zuv z uv ¯ z + ρ2uv z ∇zuv ¯ z = 0.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Thus if t = 0 is a critical point of the energy function, then (14) is equal to zero for any harmonic Beltrami differential, which implies that

• u∗(ρ2dvd¯

v) 2,0 = ρ2uv

z uv ¯ z dz2 = 0,

i.e. the harmonic map u for t = 0 is weakly conformal, and so u is ± holomorphic. Therefore, Proposition t0 ∈ T is a critical point of energy function if and only if the associated harmonic map ut0 is ± holomorphic. The above proposition was contained in Sacks-Uhlenbeck by considering the harmonic maps from Riemann surfaces to a general Riemannian manifold.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

The second derivative

For the pullback complexified tangent bundle u∗TCS = u∗K ∗

S ⊕ u∗K ∗ S, it can be equipped with the following

pointwise Hermitian inner product

• f1

∂ ∂v + f2 ∂ ∂¯ v , e1 ∂ ∂v + e2 ∂ ∂¯ v

• := ρ2(f1e1 + f2e2),

(15) and the global inner product is defined by ·, · =

• Σ

(·, ·)λ2 i 2dz ∧ d¯ z. (16) We denote ∇u∗TCS⊗KΣ

z ∂u ∂z by

∇u∗TCS⊗KΣ

∂/∂z

(∂u ∂z ⊗ dz) =: (∇u∗TCS⊗KΣ

z

∂u ∂z ) ⊗ dz.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Then the second derivative of energy function is given by Theorem The second derivative of energy function is given by 1 2 d2E(t) dt2 |t=0 =

• Σ

|du|2 |q|2 λ4 dµg0 −

• J

∂u ∂t

• , ∂u

∂t

• ,

(17) where J( ∂u

∂t ) = −Re( 2¯ q λ4

0 ∇u∗TCS⊗KΣ

z ∂u ∂z ) and

J := − 1 λ2 ∇z∇¯

z − 1

λ2 R(•, ∂u ∂z )∂u ∂¯ z (18) J is real, self-adjoint, semi-positive.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Proposition For any solution V ∈ A0(Σ, u∗TCS) of the equation J(V) = −Re        2¯ q λ4 ∇u∗TCS⊗KΣ

z

∂u ∂z        , (19) we have J(V), V =

• J

∂u

∂t

• , ∂u

∂t

• .

Corollary The second derivative of energy function is given by 1 2 d2E(t) dt2 |t=0 =

• Σ

|du|2 |q|2 λ4 dµg0 − J (V) , V , (20) where V ∈ A0(Σ, u∗TCS) is a solution of (19).

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Convexity of energy function

Denote by ∇0,1 the (0, 1)-part of the connection ∇ on u∗TCS. We define a Hermitian pointwise inner product on the line bundle KΣ by (d¯ z, d¯ z) := 1 λ2 . There is an induced inner product ·, · on the space A0,1(Σ, u∗TCS). Denote by (∇0,1)∗ the adjoint operator of ∇0,1 with respect to ·, ·. Then the action on the space A0,1(Σ, u∗TCS), (∇0,1)∗ is given by, , for example, (∇0,1)∗(fd¯ z ⊗ ∂ ∂v ) = − 1 λ2 ∇zf ⊗ ∂ ∂v . Denote the Hodge-Laplacian of ∇0,1 by ∆0,1 := ∇0,1(∇0,1)∗ + (∇0,1)∗∇0,1.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

In terms of ∆0,1 the Jacobi operator J on smooth sections of u∗TCS is J = ∆0,1 + R, (21) where R(•) := − 1

λ2

0 R(•, ∂u

∂z ) ∂u ∂¯ z , R is semi-positive. Denote by

µ = ¯ q λ2 d¯ z ⊗ ∂ ∂z the harmonic Beltrami differential. Then iµdu = ¯ q λ2 d¯ zi ∂

∂z (du) = ¯

q λ2 d¯ z ⊗ ∂u ∂z , and (∇0,1)∗iµdu = − 1 λ2 ∇z( ¯ q λ2 ∂u ∂z ) = − ¯ q λ4 ∇u∗TCS⊗KΣ

z

∂u ∂z . (22)

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Proposition The second derivative of energy function at t = 0 satisfies d2E(t) dt2 |t=0 ≥ 4

• H(iµdu)2 − Re(∇0,1)∗iµdu, J−1((∇0,1)∗iµdu)
• ,

with the equality if and only if R(J−1(∇0,1)∗iµdu) = 0, where H denotes the harmonic projection to the space Ker∆0,1 If t = 0 is a critical point, then R(J−1(∇0,1)∗iµdu) = (∇0,1)∗iµdu, J−1((∇0,1)∗iµdu) = 0 we have d2E(t) dt2 |t=0 = 4H(iµdu)2 ≥ 0, (23) i.e. the energy is convex at the critical point t0 ∈ T .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

If moreover, dut0 is never zero, then we can prove H(iµdu) 0. Thus Theorem If t0 ∈ T is a critical point of energy function, then the energy fucntion is convex at t0. If moreover, dut0 is never zero, then the energy function is strictly convex at t0.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Three key points: If u0 : Σ → S is a covering map, then u0 is a surjective map with deg u0 0, and for any p ∈ Σ, the induced homomorphism (u0)∗ : π1(Σ, p) → π1(S, u0(p)) is injective. By Schoen-Yau and Sacks-Uhlenbeck, the energy function E(t) is proper. Thus there exists a critical point t0 ∈ T such that E(t) ≥ E(t0) for all t ∈ T . For covering map, one has χ(Σ) = deg u0 · χ(S) on the

• ther hand, by Riemann-Hurwitz formula

χ(Σ) = deg ut · χ(S) −

• (di − 1) = deg u0 · χ(S) −
• (di − 1),

where di ≥ 1 is the ramification index. Which implies that di = 1, so ∂ut

∂z is never zero, so is dut.

Since E(t) is a Morse function with only index zero and T is connected, by a Morse argument E(t) has a unique critical point t0 ∈ T .

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Branched coverings

Indeed, we can give a direct proof for the uniqueness of the critical point for energy function when the reference map is a covering map Σ → S. u0 : Σ → S with deg u0 > 0, is homotopic to b ◦ p where p is a pinch map, b a branched covering. (A. Edmonds) A branched covering is a covering map except on a finite set p around which it is the form of z → zn, n > 1. Here n is called a local degree. A branched covering is simple if for each y ∈ S, u−1(y) consists of all regular points except possibly one singular point of local degree 2. Any branched covering is homotopic to a simple branched covering.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Theorem (Kim-Wan) If u : Σ → S is a covering map, E(t) : T (Σ) → R has a unique critical point, and the Hopf differential map Φ : T (S) → QD((Σ, g)), h → (u∗

hh)2,0 is injective where

uh : (Σ, g) → (S, h) is a unique harmonic map homotopic to u. If u is a non-simple branched covering, E(t) has at least two critical points. Nonetheless, the Hopf differential map Φ : T (S) → QD((Σ, g)) is injective when g = [u∗h] for some hyperbolic metric h on S.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Remark

If u0 = Id is identity map, the above Corollary was proved by Tromba. The energy density satisfies 1

2|du|2 ≥ 1 for any

harmonic map u homotopic to identity, with the equality if and only if u is identity (M. Wolf). Both of the above results were proved essentially using Schoen-Yau or Sampson that these harmonic maps are orientation preserving diffeomorphisms. A Fuchsian representation of π1(Σ) in PSL(n, R) is a representation which factors through the irreducible representation of PSL(2, R) in PSL(n, R) and a cocompact representation of π1(Σ) in PSL(2, R). A Hitchin representation is a representation which may be deformed to a Fuchsian representation.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

The space of Hitchin representation is denoted by RepH(π1(Σ), PSL(n, R)) and is called Hitchin component. Hitchin gives explicit parametrisations of Hitchin components. Namely, given a choice of a complex structure J over a given compact surface Σ, he produces a homeomorphism HJ : Q(2, J) ⊕ · · · ⊕ Q(n, J) → RepH(π1(Σ), PSL(n, R)), where Q(p, J) denotes the space of holomorphic p-differentials

• n Riemann surface (Σ, J). For each J and ρ, there exists a

unique (up to isometries) ρ-equivariant harmonic map f : Σ → SL(n, R)/SO(n), and thus obtain an energy function eρ(J) on Teichm¨ uller space. In particular, the ρ-equivariant harmonic map f is always an immersion.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Note that HJ is not mapping class group invariant, so Labourie defines an equivariant Hitchin map by H : E(n) → RepH(π1(Σ), PSL(n, R)), (J, ω) → HJ(0, ω), where E(n) is the vector bundle over Teichm¨ uller space whose fibre above the (isotopy class of the) complex structure J is E(n)

J

:= Q(3, J) ⊕ · · · ⊕ Q(n, J). The Hitchin map is surjective and eρ(J) is proper. Labourie conjecture the Hitchin map is a homeomorphism. which is also equivalent to Labourie conjecture The energy function eρ(J) has a unique critical point for any Hitchin representation ρ.

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Teichm¨ uller space Energy of harmonic maps Mapping class group invariant Kahler metrics on moduli spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S Harmonic maps for branched coverings and general remarks

Thanks for your attention!

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