Table of Contents

Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

Volume 1: Teichmüller Theory

by John H. Hubbard

ISBN: 9780971576629
459 pages, hardcover, smythe-sewn binding

2006, $85

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Foreword by William Thurston xi

Foreword to volume 1 by Clifford Earle xv

Preface

Chapter 1 The uniformization theorem

1.1 Two statements of the uniformization theorem 1
1.2 Subharmonic and harmonic functions 3
1.3 Rado's theorem 6
1.4 An exhaustion of X 10
1.5 Green's functions 14
1.6 Simply connected compact pieces 15
1.7 Proof of Theorem 1.1.2 16
1.8 A first classification of Riemann surfaces 17

Chapter 2: Plane hyperbolic geometry

2.1 The hyperbolic metric 23
2.2 Curvature of conformal metrics 31
2.3 Canoeing in the hyperbolic plane 37
2.4 The hyperboloid model and hyperbolic trignometry 48
2.5 Topological vector spaces

Chapter 3: Hyperbolic geometry of Riemann surfaces

3.1 Fuchsian groups 59
3.2 The classification of annuli 62
3.3 The hyperbolic metric on a hyperbolic Riemann surface 69
3.4 Limit sets and the convex core of a hyperbolic Riemann surface 75
3.5 Trousers 78
3.6 Trouser decomposition 84
3.7 Limit sets and ideal boundaries 87
3.8 The collaring theorem 89
3.9 Fundamental domains 95

Chapter 4: Quasiconformal maps and the mapping theorem

4.1 Two analytic definitions 111
4.2 Sobolev spaces and the Jacobian formula 117
4.3 Annuli and quasiconformal maps 124
4.4 Normal families of quasiconformal mappings 129
4.5 Geometric characterizations of quasiconformal maps 134
4.6 The mapping theorem 149
4.7 Dependence on parameters 153
4.8 Beltrami forms and complex structures 157
4.9 Boundary values of quasiconformal maps 170

Chapter 5: Preliminaries to Teichmüller theory

5.1 The Douady-Earle extension 184
5.2 Holomorphic motions and Slodkowski's theorem 194
5.3 Teichmüller extremal maps 207
5.4 Spaces of quadratic differentials 220

Chapter 6: Teichmüller spaces

6.1 Quasiconformal surfaces 235
6.2 Families of Riemann surfaces 239
6.3 The Schwarzian derivative 245
6.4 Teichmüller spaces 254
6.5 Analytic structure of Teichmüller spaces 262
6.6 Tangent spaces and Finsler metrics 266
6.7 Teichmüller spaces are contractible 273
6.8 The universal curve and the universal property 274
6.9 The Bers fiber space 284
6.10 Teichmüller spaces 287
6.11 Analytic structure of Teichmüller spaces 290
6.12 Simultaneous uniformization and quasi-Fuchsian groups 293

Chapter 7: Geometry of finite-dimensional Teichmüller spaces

7.1 Finite-dimensional Teichmüller spaces 299
7.2 Teichmüller's theorem 301
7.3 The Mumford compactness theorem 302
7.4 Royden's theorem on automorphisms of Teichmüller spaces 307
7.5 Sections of the universal Teichmüller curve 317
7.6 Fenchel-Nielsen coordinates on Teichmüller space 320
7.7 The Petersson-Weil metric 328
7.8 Wolpert's theorem 333

Appendix A

A.1 Partitions of Unity 339
A.2 Dehn twists 342
A.3 Riemann-Hurwitz 349
A.4 Almost-complex structures in higher dimensions 353
A.5 Holomorphic functions on Banach spaces and
Banach manifolds 359
A.6 Compact perturbations 371
A.7 Sheaves and cohomology 382
A.8 The Cartan-Serre theorem 402
A.9 Serre duality 407
A.10 The Riemann-Roch theorem for Riemann surfaces 413
A.11 Weierstrass points 422

Appendix B

B.1 Glossary 428
B.2 Bibliography 441

Index 446

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