An Introduction to Teichmüller Spaces

ISBN-13: 9784431681762 / Angielski / Miękka / 2011 / 279 str.

Yoichi Imayoshi; Masahiko Taniguchi

Opis

This is the English translation of our introductory book on Teichmiiller space written in Japanese. We have taken advantage of the opportunity afforded by this translation to correct some minor errors in the original text, and to include several new related topics as additional sections or subsections. Among other things, we add the construction of Thurston's compactification of the Teichmiiller space in 4 of Chapter 3, and the Thurston and Bers theory on the classification of Teichmiiller modular transformations in 5 of Chapter 6. We also include a sketch of the proof of celebrated theorems of Royden ( 4 of Chapter 6), connection between Teichmiiller theory and deformation theory of the complex structures of Riemann surfaces due to Kodaira and Spencer ( 2.4 of Chapter 7), and a derivation of negativity of curvatures of the Teichmiiller space with respect to the Weil-Petersson metric ( 3.4 of Chapter 7). Further, we indicate how to verify that the compactified moduli space of a compact Riemann surface, constructed in Appendix B, is actually compact. We include several new references chiefly related to added parts. Other than these, the main body of the text is unchanged. Professors Tadashi Kuroda and Kotaro Oikawa kindly read the manuscript of the English edition and gave much valuable advice. We also record our gratitude to all friends and colleagues, especially Sadayoshi Kojima, Makoto Masumoto, Toshiyuki Sugawa, and Harumi Tanigawa, who rendered us much help by reading various parts of the manuscript.

Spis treści

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.