Complex Analysis on Teichm uller space Hideki Miyachi Kanazawa - - PowerPoint PPT Presentation

Complex Analysis on Teichm¨ uller space

Hideki Miyachi

Kanazawa University

Teichm¨ uller Theory: Classical, Higher, Super and Quantum CIRM, Luminy (Online conference, October 5th-9th, 2020) (October 8th, 2020)

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1 Motivation 2 Notation 3 Poisson integral formula 4 Bergman kernel 5 Recent progress

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Section 1 Motivation

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Motivation

▸ Let Σg be a closed oriented surface of genus g (≥ 2). ▸ Teichm¨ uller space Tg of Σg is the moduli space of marked Riemann surfaces (I will explain later). ▸ Teichm¨ uller space Tg is realized as a bounded hyperconvex domain in C3g−3, called the Bers slice (Bers 1961, Krushkal 1991).

Problem 1 (Develop the classical Teichm¨ uller theory)

Study and develop the function theory and the pluripotential theory on Teichm¨ uller space.

Figure: Bers slice: Courtesy of Prof. Yasushi Yamashita (Nara)

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Motivation

▸ Holomorphic maps into Teichm¨ uller space Tg is understood as holomorphic families of Riemann surfaces of genus g. ⇒ Holomorphic functions from Teichm¨ uller space Tg stands for holomorphic invariants for holomorphic families of Riemann surfaces or Kleinian groups etc., like

• Period matrices
• Trace functions

Aim of this research

Study such holomorphic invariants from a higher perspective, by applying many powerful results in the function theory of several complex variables, and the pluripotential theory. ▸ This is a challenging theme! We translate (understand) theorems in the function theory of several complex variable into the language of Teichm¨ uller theory (next slide) ▸ Recently, we have more informations in Teichm¨ uller theory than 60’s!

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Motivation

Our naive idea for developments of the complex analytic theory of Teichm¨ uller space is to make an dictionary (interpretations) for interacting with several fields.

Teichm¨ uller theory Geometry & (Complex) Analysis Bdd domain, hyperconvex (Bers, Krushkal) Polynomially convex (Shiga) Teichm¨ uller space Contractible, Kobayashi complete (Teichm¨ uller, Royden) Contractible K¨ ahler mfld (Jost-Yau, with nice MCG action Daskalopoulos-Mese) Mapping class group Covering Transformation (Teichm¨ uller) Holomorphic automorphism (Royden) Teichm¨ uller metric Kobayashi-Finsler metric (Royden) Teichm¨ uller distance Kobayashi distance (Royden) Pluricomplex Green function (Krushkal, M) Log of Extremal length Horofunction (Liu-Su) Masur-Veech measure

• Inv. meas. on cotangent space

(Masur, Veech) Ratio of Extremal lengths Poisson kernel (M, Today) Thurston measure Pluriharmonic measure (M, Today) ABEM measure ≍ Bergman kernel (M-G.Hu, Today) ⋯ (To be continued)

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Motivation

Our naive idea for developments of the complex analytic theory of Teichm¨ uller space is to make an dictionary (interpretations) for interacting with several fields.

Teichm¨ uller theory Geometry & (Complex) Analysis Bdd domain, hyperconvex (Bers, Krushkal) Polynomially convex (Shiga) Teichm¨ uller space Contractible, Kobayashi complete (Teichm¨ uller, Royden) Contractible K¨ ahler mfld (Jost-Yau, with nice MCG action Daskalopoulos-Mese) Mapping class group Covering Transformation (Teichm¨ uller) Holomorphic automorphism (Royden) Teichm¨ uller metric Kobayashi-Finsler metric (Royden) Teichm¨ uller distance Kobayashi distance (Royden) Pluricomplex Green function (Krushkal, M) Log of Extremal length Horofunction (Liu-Su) Masur-Veech measure

• Inv. meas. on cotangent space

(Masur, Veech) Ratio of Extremal lengths Poisson kernel (M, Today) Thurston measure Pluriharmonic measure (M, Today) ABEM measure ≍ Bergman kernel (M-G.Hu, Today) ⋯ (To be continued)

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Motivation

One of the important tools in the (pluri)potential theory in the case of dimension 1 is the Poisson integral formula:

Theorem 1 (Poisson integral formula)

Let u be a harmonic function on D = {∣z∣ < 1} which is continuous on D. Then, u(z) = ∫

2π

u(eiθ) 1 − ∣z∣2 ∣eiθ − z∣2 dθ 2π (z ∈ D). The measure ωD,x = 1 − ∣z∣2 ∣eiθ − z∣2 dθ 2π is called the harmonic measure on D at z ∈ D.

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Motivation : Demailly’s work

A domain Ω ⊂ CN is said to be hyperconvex if it admits a bounded (from above) PSH-exhaustion.

Theorem 2 (Demailly 1987)

A bounded hyperconvex domain Ω ⊂ CN admits a unique pluricomplex green function gΩ on Ω and the pluriharmonic measure {ωΩ,x}x∈Ω. The pluricomplex Green function on Ω is defined by gΩ(x,y) = sup{v(y) ∣ v ∈ PSH(Ω), v ≤ 0, v(y) = log ∥y − x∥ + O(1)}. The pluriharmonic measure ωΩ,x is a Borel measure supported on ∂Ω satisfying V (x) = ∫∂Ω V dωΩ,x − ∫Ω ddcV ∧ ∣uz∣(ddcuz)N−1 for V ∈ PSH(Ω) ∩ C0(Ω) where ux = (2π)−1gΩ(x,⋅).

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Motivation : Demailly’s work

A domain Ω ⊂ CN is said to be hyperconvex if it admits a bounded (from above) PSH-exhaustion.

Theorem 2 (Demailly 1987)

A bounded hyperconvex domain Ω ⊂ CN admits a unique pluricomplex green function gΩ on Ω and the pluriharmonic measure {ωΩ,x}x∈Ω. The pluricomplex Green function on Ω is defined by gΩ(x,y) = sup{v(y) ∣ v ∈ PSH(Ω), v ≤ 0, v(y) = log ∥y − x∥ + O(1)}. The pluriharmonic measure ωΩ,x is a Borel measure supported on ∂Ω satisfying V (x) = ∫∂Ω V dωΩ,x − ∫Ω ddcV ∧ ∣uz∣(ddcuz)N−1 for V ∈ PH(Ω) ∩ C0(Ω) where ux = (2π)−1gΩ(x,⋅).

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Motivation : Bergman kernel

▸ Let M be an n-dimensional complex manifold and OL2(M) the space of L2-holomorphic n-forms with the inner product in2 2n ∫M f1 ∧ f2 (f1,f2 ∈ OL2(M)). ▸ Let {fj(z)dZ}∞

j=1 (dZ = dz1 ∧ ⋯ ∧ dzn) be a complete orthonormal system on

OL2(M) and set KM(z,w)dZ ⊗ dW =

∞

∑

j=1

fj(z)fj(w)dZ ⊗ dW, which is the reproducing kernel on OL2(M). ▸ The Bergman kernel is defined by KM = KM(z,w)dZ ⊗ dZ ▸ The Bergman Kernel is a useful biholomorphic invariant in complex geometry.

• C.Fefferemen analyzed the boundary behavior of the Bergman kernel on

strongly pseudoconvex domains with C∞-boundaries, and proved that any biholomorphic map between strongly pseudoconvex domain in Cn extends smoothly to a diffeomorphism between their closures.

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Motivation

▸ (Recall) Teichm¨ uller space is realized as a bounded hyperconvex domain, called the Bers slice (Bers 1961, Krushkal 1991). ▸ (Recall) Teichm¨ uller space is the moduli space of marked Riemann surfaces.

Problem 2 (Today’s topic)

1 Description of the pluricomplex Green function and the pluriharmonic

measure on Teichm¨ uller space in terms of the conformal (biholomorphic) invariants of Riemann surfaces.

2 Estimation of the Bergman kernel in terms of the invariant in the Teichm¨

uller theory (joint work with Guangming Hu).

Figure: Bers slice: Courtesy of Prof. Yasushi Yamashita (Nara)

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Section 2 Notation

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Teichm¨ uller space

▸ A marked Riemann surface of genus g is a pair (X,f) of a Riemann surface X of genus g and an orientation preserving homeomorphism f∶Σg → X. ▸ Two marked Riemann surfaces (X1,f1) and (X2,f2) are said to be Teichm¨ uller equivalent if there is a biholomorphism h∶X1 → X2 such that h ○ f1 is homotopic to f2: Σg

f1

• f2
• X1

h

• X2

▸ The Teichm¨ uller space Tg of genus g consists of the Teichm¨ uller equivalence classes of marked Riemann surfaces of genus g.

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Teichm¨ uller theory : Extremal length

▸ Let S = Sg be the set of homotopy classes of non-trivial simple closed curves

• n Σg.

▸ Let α ∈ S and x = (X,f) ∈ Tg, we define the extremal length of α on x by Extx(α) = sup

ρ

inf

α′∼f(α)∫α′ ρ(z)∣dz∣

• X

ρ(z)2dxdy where ρ = ρ(z)∣dz∣ rums all conformal (measurable) metrics on X.

Remark 1

There is a unique quadratic differential qα,x = qα,x(z)dz2 ∈ H0(X,K⊗2

X ) such that

Extx(α) = ∫X ∣qα,x(z)∣dxdy (z = x + iy). The differential qα,x is called the Jenkins-Strebel differential on x = (X,f) for α.

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Teichm¨ uller distance

▸ The Teichm¨ uller distance dT is defined by dT (x,y) = 1 2 log sup

α∈S

Extx(α) Exty(α) (x,y ∈ Tg) (Kerckhoff formula, Kerckhoff 1980). ▸ Teichm¨ uller distance coincides with the Kobayashi distance (Royden 1971). ▸ Teichm¨ uller distance is unique geodesic (Teichm¨ uller 1940). ▸ Teichm¨ uller distance is a complete Finsler distance, and the Finsler metric, called the Teichm¨ uller metric, is of C1 (Royden 1971). ▸ When g = 1, Teichm¨ uller distance coincides with the Poincar´ e distance on H

• f curvature −4.

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Theorem 3 (M, 2017, 2019)

Extremal length function is “nice”: The minus of the reciprocal Tg ∋ x ↦ uF (x) = − 1 Extx(F) is plurisubharmonic for F ∈ ML − {0}. Furthermore, (∂∂uF )3g−4 ≠ 0 and (∂∂uF )3g−3 = 0 when F is essentially complete (i.e. Σg ∖ ∣F∣ consists of ideal triangles). ▸ From Bedford-Kalka’s results on holomorphic foliations (Bedford-Kalka, 1977), the last statement implies that there is a foliation in Tg by Riemann surfaces. ▸ In this case, the leaves are nothing but the Teichm¨ uller disks associated to quadratic differentials whose vertial foliations are F. ▸ When F = α ∈ S (in this case, F is not essentially complete), we also have a foliation by one-dimensional holomorphic disks, Teichm¨ uller disks (Marden-Masur 1975). On each leaf, uα is harmonic.

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Section 3 Poisson integral formula

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Teichm¨ uller space (g = 1)

▸ Teichm¨ uller space T1 of genus 1 is identified with the upper half plane H via the period: ▸ The Teichm¨ uller distance on T1 coincides with the Poincar´ e distance on H of curvature −4. ▸ The Green function gT1 on T1 satisfies gT1(x,y) = log tanhdT (x,y) (x,y ∈ T1)

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Simple closed curves and Extremal length (g = 1)

▸ When g = 1, S is identified with ˆ Q = Q ∪ {∞} (∞ = 1/0 = −1/0), and simple closed curves ˆ Q is understood as a subset of ∂H by the pinching deformaiton. ▸ The simple closed curve p + qτ on Xτ = C/Z ⊕ Zτ corresponds to −p/q ∈ ˆ Q. ▸ For p/q ∈ ˆ Q = S and τ ∈ H = T1, Extτ(p/q) = ∣ − p + qτ∣2 Im(τ) , qp/q,τ = −(−p + qτ)2 Im(τ)2 dz2. ▸ In particular, when p/q = 1/0 = ∞ ∈ ˆ Q, Extτ(1/0) = 1 Im(τ). Hence, {up/q = Const.} is thought to be a horocircle.

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Harmonic measure is the visual angle measure (g = 1)

Geometrically, the harmonic measure on H ≅ T1 is the pushforward measure of the “visual angle” (with total angle 1): where Θτ is the visual angle measure at τ ∈ H defined on the infinitesimal circle.

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Poisson kernel in Teichm¨ uller theory (Case g = 1)

▸ Notice that the (pluri)harmonic measure at τ ∈ H = T1 is ωT1,τ(E) = ∫E ( η(t2 + 1) (ξ − t)2 + η2 )dΘ√

−1(t)

(τ = ξ + √ −1η) ▸ The extremal length of the p/q-curve on τ = ξ + √ −1η ∈ H = T1 is Extτ(p/q) = ∣ − p + qτ∣2 Im(τ) = (qξ − p)2 + q2η2 η Hence Ext√

−1(p/q)

Extτ(p/q) = (q ⋅ 0 − p)2 + q2 ⋅ 12 η (qξ − p)2 + q2η2 η = η((p/q)2 + 1) (ξ − (p/q))2 + η2

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Poisson integral formula in Teichm¨ uller theory (g = 1)

Theorem 4 (Green function and Poisson integral formula(Case g = 1))

▸ (Green function) (pluricomplex) Green function gT1 on T1 satisfies gT1(x,y) = log tanhdT (x,y) ▸ (Poisson kernel) For τ ∈ T1 = H, the quotient T1 × S = T1 × (Q ∪ {∞}) ∋ (τ,p/q) ↦ P( √ −1,τ,p/q) = Ext√

−1(p/q)

Extτ(p/q) extends continuously on T1 × (Infinitesimal circle), where (Infinitesimal circle) ≅ ∂H = R ∪ {∞}. ▸ (Poisson integral formula) Let u be a harmonic function on T1 (= H) which is continuous on the closure (= H). Then, u(τ) = ∫∂T1 u(t)( Ext√

−1(t)

Extτ(t) )dΘ√

−1(t).

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Higher genus case : Generalization?

▸ The Teichm¨ uller distance dT on Tg is the Kobayashi distance (Roydan 1971). ▸ The pluricomplex Green function satisfies gTg(x,y) = log tanhdT (x,y). (Krushkal 1992, M 2019). ▸ Our task : How do we describe the pluriharmonic measure? The case of g = 1: ωT1,τ(E) = ∫E ( Ext√

−1(t)

Extτ(t) ) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ Conformal invariant dΘ√

−1(t)

ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ Pushforward of Vis.ang.meas. for E ⊂ ∂T1 = ∂H ≅ (infinitesimal circle).

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Higher genus case : Generalization?

✓ ✏ dωT1,τ = ( Ext√

−1(⋅)

Extτ(⋅) )dΘ√

−1

• n ∂T1 = ∂H ≅ (infinitesimal circle)

✒ ✑ ▸ (Infinitesimal sphere ) There is a completion PMF of S which represents the infinitesimal sphere (Thurston, Hubbard-Masur 1978). ▸ (Visual angle measure ) For x0 ∈ Tg, there is a (unique) probability measure ˆ µx0

T h on PMF (Thurston).

▸ (Poisson kernel ) For x0 ∈ Tg, Tg × S ∋ (x,α) ↦ Extx0(α) Extx(α) extends continuously on Tg × PMF (Kerckhoff 1970). ▸ (Correspondence ) There is no (canonical) identification between the visual sphere PMF and the (Bers) boundary ∂Tg.

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Higher genus case : Generalization?

▸ (Correspondence ) There are a measurable subset PMFue ⊂ PMF with ˆ µx0

T h(PMFue) = 1 and a canonical identification (Ending Lamination

Theorem, Brock-Canary-Minsky). Φx0∶PMFue

≅

• → ∂ueTg ⊂ ∂Tg.

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Main theorem : Poisson integral formula

Actually, we have a generalization.

Theorem 5 (Poisson integral formula)

Let V be a continuous function on the (Bers) closure Tg which is pluriharmonic

• n Tg. Then

V (x) = ∫∂Tg V (φ)P(x0,x,φ)dµB

x0(φ)

P(x,y,φ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (Extx(Fϕ) Exty(Fϕ))

3g−3

(φ ∈ ∂ueTg) 1 (otherwise)

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Main theorem : Poisson integral formula

Actually, we have a generalization.

Theorem 5 (Poisson integral formula)

Let V be a continuous function on the (Bers) closure Tg which is pluriharmonic

• n Tg. Then

V (x) = ∫∂Tg V (φ)P(x0,x,φ) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

• Conf. Inv.

dµB

x0(φ)

ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ Pushforward of Vis.ang.meas. P(x,y,φ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (Extx(Fϕ) Exty(Fϕ))

3g−3

(φ ∈ ∂ueTg) 1 (otherwise)

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Main theorem : Poisson integral formula

Corollary 3 (Poisson integral formula (Topological description))

Let V be a continuous function on the (Bers) closure Tg which is pluriharmonic

• n Tg. Then

V (x) = ∫PMF ˆ V ([F])(Extx0(F) Extx(F) )

3g−3

ˆ µx0

T h([F])

= ∫PMF ˆ V ([F])e(6g−6)β(x0,x,[F ])ˆ µx0

T h([F])

where ˆ V = V ○ Φx0 and we identify SMFx0 ≅ PMF. Thus, the ratio of the extremal lengths is the Poisson kernel, and the Thurston measure is the pluriharmonic measure. Here, we define the horofunction (Busemann function, cocycle function) by β(x,y,[F]) = 1 2(log Extx(F) − log Exty(F)) (following Lixin Liu and Weixu Su).

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Application : Representation of hol. quad. differentials

Let V be a pluriharmonic function on Tg which is continuous on the Bers closure. Then, V (x) = ∫PMF ˆ V ([F])(Extx0(F) Extx(F) )

3g−3

dˆ µx0

T h([F]),

(∂V )x = (3g − 3)∫PMF ˆ V ([F])(Extx0(F) Extx(F) )

3g−3 qF,x

∥qF,x∥dˆ µx0

T h([F]),

because of the Gardiner formula: (∂Ext⋅(F))x = −qF,x (x ∈ Tg) and ∥qF,x∥ = Extx(F), where qF,x is the Hubbard-Masur differential for F ∈ ML at x ∈ Tg.

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Application : Representation of hol. quad. differentials

Let V be a pluriharmonic function on Tg which is continuous on the Bers closure. Then, V (x) = ∫PMF ˆ V ([F])(Extx0(F) Extx(F) )

3g−3

dˆ µx0

T h([F]),

(∂V )x = (3g − 3)∫PMF ˆ V ([F])(Extx0(F) Extx(F) )

3g−3 qF,x

∥qF,x∥dˆ µx0

T h([F]),

because of the Gardiner formula: (∂Ext⋅(F))x = −qF,x (x ∈ Tg) and ∥qF,x∥ = Extx(F), where qF,x is the Hubbard-Masur differential for F ∈ ML at x ∈ Tg.

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Application : Representation of hol. quad. differentials

For γ ∈ π1(Σg), let Θγ,x be a holomorphic quadratic differential with dhLengγ[v] = Re∫M µΘγ,x for v = [µ] ∈ TxTg at x = (M,f) ∈ Tg (Gardiner, Wolpert).

Theorem 4

Θγ,x = 3g − 3 2sinh(hLengγ(x)) ∫PMFmf tr2(ρϕF,x(γ)) qF,x ∥qF,x∥dˆ µx0

T h([F])

where φF,x ∈ ∂T B

x0 ([F] ∈ PMFmf) is the totally degenerate group whose ending

lamination is the support of F.

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Very rough idea of the proof of PIF

We apply the Demailly’s theory. Indeed, we calculate the Poisson kernel, and the pluriharmonic measure by applying several results from the function theory of several complex variables (Demailly, Bedford-Taylor, ...).

1 For Poisson kernel,

→ Use Extremal length geometry to show ( gTg(y, z) gTg(x, z))

3g−3

→ (Extx(Fϕ) Exty(Fϕ))

3g−3

(= P(x, y, ϕ)) as z → ϕ ∈ ∂ueTg which gives the presentation of the Poisson kernel (Demailly).

2 For the Pluriharmonic measure, 1 Show ωTg,x is absolutely continuous with respect to µB

x ;

2 Calculate the transformation law of the Pushforward measures {µB

x }x∈Tg to

show that the Radon-Nikodym derivative f = dωTg,x/dµB

x satisfies that

f ○ Φx0 is an MCG-invariant measurable function on PMF;

3 Since the action of the MCG on PMF is ergodic, f is constant; 4 Since ωTg,x and µB

x are probability measure, f ≡ 1.

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Section 4 Bergman kernel

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Bergman kernel on Teichm¨ uller space

▸ Let π∶Qg → Tg be the cotangent bundle of Tg we define the measure on ˜ µMV

• n Qg by

˜ µMV(E) = (µT h × µT h)(H(E) × V(E)), where E ⊂ Qg, H(q) and H(q) are the horizontal and vertical foliations of q ∈ Qg, and µT h is the Thurston measure. ▸ Let π∶Q0

g → Tg be the unit sphere bundle and define a measure µMV on Q0 g by

µMV(E) = ˜ µMV({tq ∣ q ∈ E,0 ≤ t ≤ 1}). We call µMV the Masur-Veech measure. This is a (unique) invariant measure for the Teichm¨ uller flow. ▸ We define a measure mMV on Tg as the push-forward measure mA

BEM = π∗(µMV).

We call mA

BEM the ABEM measure on Tg (Athreya - Bufetov - Eskin -

Mirzakhani) here. This is invariant under the action of the mapping class group.

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Comparison between KTg and mABEM

▸ Recall that the Bergman kernel KTg = KTg(z)dZ ⊗ dZ is a non-negative 2(3g − 3)-form on Tg, hence it can be compared with volume forms.

Theorem 5 (M - Guangming Hu)

There are positive constants C1 and C2 depending only on g such that C1mA

BEM ≤ KTg ≤ C2mA BEM

• n Tg.

Since mA

BEM(Mg) < ∞ (Masur), we have

Corollary 6 (Integral is finite on the moduli space)

We have ∫Mg KTg < ∞.

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Idea of the proof

▸ A crucial part : We compare the Bergman kernel with the Busemann measure µB which is defined by dµB(x) = ϵ6g−6 VE(IT (x))dVE (x ∈ Tg), where we identify Tg as a bounded domain (Bers slice with basepoint x), and VE is the Euclidean measure on Tg ⊂ C3g−3 and TxTg ≅ C3g−3 and IT (x) is the Kobayashi-Teichm¨ uller indicatrix IT (x) = {v ∈ TxTg ∣ FT (v) ≤ 1} (FT is the Teichm¨ uller metric). ϵ6g−6 is the Euclidean volume of the unit ball in C3g−3 ≅ R6g−6. ▸ The comparison µB ≍ KTg is proved by applying the argument by Z.B locki for convex domains. ▸ µB ≍ mA

BEM is already observed by S.Dowdall, M.Duchin and H.Masur.

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Section 5 Recent progress

Hideki Miyachi (Kanazawa Univ.) Complex Analysis

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Recent pregress : Bounded holomorphic functions

▸ (“Fatou type theorem” is true) Any bounded holomorphic function admits radial limits on the Bers boundary almost everywhere with respect to the pluricomplex harmonic measure. ▸ The following is an open problem for me:

Question 1 (I wish to prove)

Are bounded holomorphic functions presented by the Poisson integral of the radial limits? ▸ If it is true, we have the following idenitification : (Bdd hol.fns) ↔ (Bdd meas.fns on PMF with some properties) ▸ This identification provides a topological description of holomorphic functions

• n Teichm¨

uller space.

• New interaction between Topology and Complex analysis in Teichm¨

uller theory!!

Hideki Miyachi (Kanazawa Univ.) Complex Analysis

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Thank you very much for your attention (^ ^)/ !!

Hideki Miyachi (Kanazawa Univ.) Complex Analysis

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