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) Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 1 / 36
1 Motivation 2 Notation 3 Poisson integral formula 4 Bergman kernel 5 Recent progress Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 2 / 36
Section 1 Motivation Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 3 / 36
Motivation ▸ Let Σ g be a closed oriented surface of genus g ( ≥ 2 ). ▸ Teichm¨ uller space T g of Σ g is the moduli space of marked Riemann surfaces (I will explain later). ▸ Teichm¨ uller space T g is realized as a bounded hyperconvex domain in C 3 g − 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) Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 4 / 36
Motivation ▸ Holomorphic maps into Teichm¨ uller space T g is understood as holomorphic families of Riemann surfaces of genus g . ⇒ Holomorphic functions from Teichm¨ uller space T g 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! Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 5 / 36
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) Covering Transformation (Teichm¨ uller) Mapping class group Holomorphic automorphism (Royden) Teichm¨ uller metric Kobayashi-Finsler metric (Royden) Kobayashi distance (Royden) Teichm¨ uller distance 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) ⋯ Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 6 / 36
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) Covering Transformation (Teichm¨ uller) Mapping class group Holomorphic automorphism (Royden) Teichm¨ uller metric Kobayashi-Finsler metric (Royden) Kobayashi distance (Royden) Teichm¨ uller distance 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) ⋯ Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 6 / 36
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 ( e iθ ) 1 − ∣ z ∣ 2 u ( z ) = ∫ ( z ∈ D ) . 2 π dθ ∣ e iθ − z ∣ 2 2 π 0 The measure ω D ,x = 1 − ∣ z ∣ 2 dθ ∣ e iθ − z ∣ 2 2 π is called the harmonic measure on D at z ∈ D . Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 7 / 36
Motivation : Demailly’s work A domain Ω ⊂ C N is said to be hyperconvex if it admits a bounded (from above) PSH-exhaustion. Theorem 2 (Demailly 1987) A bounded hyperconvex domain Ω ⊂ C N admits a unique pluricomplex green function g Ω on Ω and the pluriharmonic measure { ω Ω ,x } x ∈ Ω . The pluricomplex Green function on Ω is defined by g Ω ( x,y ) = sup { v ( y ) ∣ v ∈ PSH ( Ω ) , v ≤ 0 , v ( y ) = log ∥ y − x ∥ + O ( 1 )} . The pluriharmonic measure ω Ω ,x is a Borel measure supported on ∂ Ω satisfying V ( x ) = ∫ ∂ Ω V dω Ω ,x − ∫ Ω dd c V ∧ ∣ u z ∣( dd c u z ) N − 1 for V ∈ PSH ( Ω ) ∩ C 0 ( Ω ) where u x = ( 2 π ) − 1 g Ω ( x, ⋅ ) . Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 8 / 36
Motivation : Demailly’s work A domain Ω ⊂ C N is said to be hyperconvex if it admits a bounded (from above) PSH-exhaustion. Theorem 2 (Demailly 1987) A bounded hyperconvex domain Ω ⊂ C N admits a unique pluricomplex green function g Ω on Ω and the pluriharmonic measure { ω Ω ,x } x ∈ Ω . The pluricomplex Green function on Ω is defined by g Ω ( x,y ) = sup { v ( y ) ∣ v ∈ PSH ( Ω ) , v ≤ 0 , v ( y ) = log ∥ y − x ∥ + O ( 1 )} . The pluriharmonic measure ω Ω ,x is a Borel measure supported on ∂ Ω satisfying V ( x ) = ∫ ∂ Ω V dω Ω ,x − ∫ Ω dd c V ∧ ∣ u z ∣( dd c u z ) N − 1 for V ∈ PH ( Ω ) ∩ C 0 ( Ω ) where u x = ( 2 π ) − 1 g Ω ( x, ⋅ ) . Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 8 / 36
Motivation : Bergman kernel ▸ Let M be an n -dimensional complex manifold and O L 2 ( M ) the space of L 2 -holomorphic n -forms with the inner product i n 2 ( f 1 ,f 2 ∈ O L 2 ( M )) . 2 n ∫ M f 1 ∧ f 2 ▸ Let { f j ( z ) dZ } ∞ j = 1 ( dZ = dz 1 ∧ ⋯ ∧ dz n ) be a complete orthonormal system on O L 2 ( M ) and set K M ( z,w ) dZ ⊗ dW = f j ( z ) f j ( w ) dZ ⊗ dW, ∞ ∑ j = 1 which is the reproducing kernel on O L 2 ( M ) . ▸ The Bergman kernel is defined by K M = K M ( 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 C n extends smoothly to a diffeomorphism between their closures. Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 9 / 36
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) Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 10 / 36
Section 2 Notation Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 11 / 36
� � � 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 ∶ Σ g → X . ▸ Two marked Riemann surfaces ( X 1 ,f 1 ) and ( X 2 ,f 2 ) are said to be Teichm¨ uller equivalent if there is a biholomorphism h ∶ X 1 → X 2 such that h ○ f 1 is homotopic to f 2 : f 1 Σ g X 1 h f 2 X 2 ▸ The Teichm¨ uller space T g of genus g consists of the Teichm¨ uller equivalence classes of marked Riemann surfaces of genus g . Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 12 / 36
Teichm¨ uller theory : Extremal length ▸ Let S = S g be the set of homotopy classes of non-trivial simple closed curves on Σ g . ▸ Let α ∈ S and x = ( X,f ) ∈ T g , we define the extremal length of α on x by α ′ ∼ f ( α ) ∫ α ′ ρ ( z )∣ dz ∣ inf Ext x ( α ) = sup ρ ( z ) 2 dxdy � ρ X where ρ = ρ ( z )∣ dz ∣ rums all conformal (measurable) metrics on X . Remark 1 There is a unique quadratic differential q α,x = q α,x ( z ) dz 2 ∈ H 0 ( X,K ⊗ 2 X ) such that Ext x ( α ) = ∫ X ∣ q α,x ( z )∣ dxdy ( z = x + iy ) . The differential q α,x is called the Jenkins-Strebel differential on x = ( X,f ) for α . Hideki Miyachi (Kanazawa Univ.) Complex Analysis Oct. 8th, 2020 13 / 36
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