An Introduction to Teichmüller Spaces

1 Teichmüller Space of Genus g.- 1.1 Riemann Surfaces.- 1.2 Teichmüller Space of Genus 1.- 1.3 Teichmüller Space of Genus g.- 1.4 Quasiconformal Mappings and Teichmüller Space.- 1.5 Complex Structures and Conformal Structures.- Notes.- 2 Frike Space.- 2.1 Uniformization Theorem.- 2.2 Universal Coverings.- 2.3 Möbius Transformations.- 2.4 Fuchsian Models.- 2.5 Fricke Space.- Notes.- 3 Hyperbolic Geometry and Fenchel-Nielsen Coordinates.- 3.1 Poincaré Metric and Hyperbolic Geometry.- 3.2 Fenchel-Nielsen Coordinates.- 3.3 Fricke-Klein Embedding.- 3.4 Thurston’s Compactification.- Notes.- 4 Quasiconformal Mappings.- 4.1 Definitions and Elementary Properties.- 4.2 Existence Theorems on Quasiconformal Mappings.- 4.3 Dependence on Beltrami Coefficients.- 4.4 Proof of Calderón-Zygmund Theorem.- Notes.- 5 Teichmüller Spaces.- 5.1 Analytic Construction of Teichmüller Spaces.- 5.2 Teichmüller Mappings and Teichmüller’s Theorerms.- 5.3 Proof of Teichmüller’s Uniqueness Theorem.- Notes.- 6 Complex Analytic Theory of Teichmüller Spaces.- 6.1 Bers’ Embedding.- 6.2 Invariance of Complex Structure of Teichmüller Space.- 6.3 Teichmüller Modular Groups.- 6.4 Royden’s Theorems.- 6.5 Classification of Teichmüller Modular Transformations.- Notes.- 7 Weil-Petersson Metric.- 7.1 Petersson Scalar Product and Bergman Projection.- 7.2 Infinitesimal Theory of Teichmüller Spaces.- 7.3 Weil-Petersson Metric.- Notes.- 8 Fenchel-Nielsen Deformations and Weil-Petersson Metric.- 8.1 Fenchel-Nielsen Deformations.- 8.2 A Variational Formula for Geodesic Length Functions.- 8.3 Wolpert’s Formula.- Notes.- Appendices.- A Classical Variations on Riemann Surfaces.- Notes.- B Compactification of the Moduli Space.- Notes.- References.- List of Symbols.