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 1 / 54

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 Teichm¨ uller space 1 Energy of harmonic maps 2 Mapping class group invariant Kahler metrics on moduli 3 spaces containing T as totally geodesic subspace Convexity of energy function for Σ → S 4 Harmonic maps for branched coverings and general 5 remarks 2 / 54

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 3 / 54

� � 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 d T (( f , X ) , ( g , Y )) = 1 2 log min h : X → Y K ( h ) where h is quasi-conformal making the diagram commute up to htpy and K ( h ) is the quasi-conformal constant. f Σ X ❄ ❄ ⑦ ❄ ⑦ ❄ ⑦ ❄ ⑦ ❄ ⑦ g ❄ ⑦ h ❄ ⑦ � ⑦ Y Teich distance d T is Finslerian (not Riemannian) 4 / 54

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 , d T ) . ( T , d T ) not Gromov hyperbolic (Masur-Wolf) T is a complex mfd of complex dim=3 g (Σ) − 3 where g (Σ) = genus of Σ . Furthermore it is Stein i.e. ∃ proper plurisubharmonic map on 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. 5 / 54

� 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 z = π − 1 ( z ) Riem. surf. w/ cx str z ∈ T loc holo coord v X π T Teichm¨ uller space of Σ , local holo coord z = ( z 1 , · · · , z 3 g − 3 ) ( z , v ) local holomorphic coordinates of X around π − 1 ( z ) T X = V ⊕ H V = < ∂ δ ∂ ∂ ∂ v >, V ∗ = < δ v = dv − a v α dz α > ∂ z α + a v ∂ v >, H = < δ z α = α 6 / 54

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 z equipped with K¨ ahler metric of K¨ ahler form √ √ v dv ∧ d ¯ v ( dv ⊗ d ¯ v − d ¯ ω X z = − 1 φ v ¯ v = − 1 φ v ¯ v ⊗ dv ) e φ = φ v ¯ v = ∂ v ∂ ¯ v φ K¨ ahler Einstein equation v ( dv ⊗ d ¯ v + d ¯ hyperbolic metric Φ z = φ v ¯ v ⊗ dv ) 7 / 54

� 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 Kodaira − Spencer � H 1 ( X z , T X z ) = H 0 ( X z , K 2 X z ) ∗ T z T Serre duality Harmonic Beltrami differentials harm . hori . lift ∂ ∂ z α + a v ∂ ∂ ∂ v ( ∂ ∂ z α + a v ∂ v ) = A v ∂ ∂ ∂ v ⊗ d ¯ ∂ v → v ∂ z α α α α ¯ v Kodaira-Spencer class 8 / 54

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 X z is locally λ ( v ) | dv | and φ, ψ ∈ T ∗ X z T quad diff, then � φ ¯ ψ < φ, ψ > = λ 2 dV X z X z or √ β dz α ∧ d ¯ z β ω WP = − 1 G α ¯ � β = � ∂ ∂ z α , ∂ ∂ ∂ ( A v v , A v ∂ v ⊗ d ¯ ∂ v ⊗ d ¯ G α ¯ ∂ z β � = v ) dV X z α ¯ β ¯ v v X z � √ A v v A v v dv ∧ d ¯ = − 1 φ v ¯ v α ¯ β ¯ v X z � = c ( φ ) α ¯ β dV X z . X z 9 / 54

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 √ √ √ β dz α ∧ d ¯ z β + − 1 ∂ ¯ v δ v ∧ δ ¯ ∂φ = − 1 c ( φ ) α ¯ − 1 φ v ¯ v β − φ v ¯ v φ α ¯ c ( φ ) α ¯ β = φ α ¯ β > 0 . v φ v ¯ v = φ − 1 α = − φ v ¯ v , φ v ¯ a v v φ α ¯ v ¯ v A v v a v v = ∂ ¯ α α ¯ v = A v v = −∇ ¯ v = − φ α ¯ A α ¯ v φ v ¯ v φ α ¯ v ¯ v ;¯ v α ¯ A α ¯ v ; v = 0 v ¯ ( − φ v ¯ v A ¯ v ∂ v ∂ ¯ β = A v v v + 1 ) c ( φ ) α ¯ β v . α ¯ ¯ 10 / 54

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) 1 Ricci curvature ≤ − 2 π ( g − 1 ) . (Tromba, Wolpert) W-P volume of moduli space T / Mod ( S g ) is calculated by Mirzakhani. Question: Are there better K¨ ahler metrics? 11 / 54

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. 12 / 54

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 || ˙ X 0 || 2 d 2 dt 2 log ℓ ( X t , g 0 ) | t 0 = 4 WP area ( X 0 ) . 3 length on X t of a random geodesic on X 0 McMullen || ˙ X 0 || 2 P = 4 || ˙ φ X || 2 WP area ( X 0 ) . 3 Pressure metric pulls back to multiple of W-P under thermodynamic embedding 13 / 54

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 ( D X ( Re θ ( c − , · )))( Y ) = � X , Y � WP , θ ( c − , · ) = 4 ∂ω M where ω M 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 .ω WP . 14 / 54

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 = ( g ij ) , { x i } ) → (Σ , h , { v } ) ∇ : ∧ l T ∗ M ⊗ u ∗ T Σ → ∧ l + 1 T ∗ M ⊗ u ∗ T Σ L 2 -inner product on ∧ l T ∗ M ⊗ u ∗ T Σ � � , � = ( , ) dvol g M �∇ , � = � , ∇ ∗ � △ = ∇ ∗ ∇ + ∇∇ ∗ Laplace operator=self adj, semi-pos elliptic operator Harmonic form △ ω = 0 ⇔ ker ∇ ∩ ker ∇ ∗ 15 / 54