Matrix
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*serious mathematics, written with the reader in mind*

459 pages, hardcover, smythe-sewn binding

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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 functions3

1.3 Rado's theorem6

1.4 An exhaustion ofX 10

1.5 Green's functions14

1.6 Simply connected compact pieces15

1.7 Proof of Theorem 1.1.216

1.8 A first classification of Riemann surfaces17

Chapter 2: Plane hyperbolic geometry

2.1 The hyperbolic metric

23

2.2 Curvature of conformal metrics31

2.3 Canoeing in the hyperbolic plane37

2.4 The hyperboloid model and hyperbolic trignometry48

2.5 Topological vector spaces

Chapter 3: Hyperbolic geometry of Riemann surfaces

3.1 Fuchsian groups

59

3.2 The classification of annuli62

3.3 The hyperbolic metric on a hyperbolic Riemann surface69

3.4 Limit sets and the convex core of a hyperbolic Riemann surface75

3.5 Trousers78

3.6 Trouser decomposition84

3.7 Limit sets and ideal boundaries87

3.8 The collaring theorem89

3.9 Fundamental domains95

Chapter 4: Quasiconformal maps and the mapping theorem

4.1 Two analytic definitions

111

4.2 Sobolev spaces and the Jacobian formula117

4.3 Annuli and quasiconformal maps124

4.4 Normal families of quasiconformal mappings129

4.5 Geometric characterizations of quasiconformal maps134

4.6 The mapping theorem149

4.7 Dependence on parameters153

4.8 Beltrami forms and complex structures157

4.9 Boundary values of quasiconformal maps170

Chapter
5: Preliminaries to Teichmüller theory

5.1 The Douady-Earle extension

184

5.2 Holomorphic motions and Slodkowski's theorem194

5.3 Teichmüller extremal maps207

5.4 Spaces of quadratic differentials220

Chapter 6: Teichmüller spaces

6.1 Quasiconformal surfaces

235

6.2 Families of Riemann surfaces239

6.3 The Schwarzian derivative245

6.4 Teichmüller spaces254

6.5 Analytic structure of Teichmüller spaces262

6.6 Tangent spaces and Finsler metrics266

6.7 Teichmüller spaces are contractible273

6.8 The universal curve and the universal property274

6.9 The Bers fiber space284

6.10 Teichmüller spaces287

6.11 Analytic structure of Teichmüller spaces290

6.12 Simultaneous uniformization and quasi-Fuchsian groups293

Chapter 7: Geometry of finite-dimensional Teichmüller spaces

7.1 Finite-dimensional Teichmüller spaces

299

7.2 Teichmüller's theorem301

7.3 The Mumford compactness theorem302

7.4 Royden's theorem on automorphisms of Teichmüller spaces307

7.5 Sections of the universal Teichmüller curve317

7.6 Fenchel-Nielsen coordinates on Teichmüller space320

7.7 The Petersson-Weil metric328

7.8 Wolpert's theorem333

Appendix A

A.1 Partitions of Unity

339

A.2 Dehn twists342

A.3 Riemann-Hurwitz349

A.4 Almost-complex structures in higher dimensions353

A.5 Holomorphic functions on Banach spaces and

Banach manifolds359

A.6 Compact perturbations371

A.7 Sheaves and cohomology382

A.8 The Cartan-Serre theorem402

A.9 Serre duality407

A.10 The Riemann-Roch theorem for Riemann surfaces413

A.11 Weierstrass points422

Appendix B

B.1 Glossary

428

B.2 Bibliography 441

Index
*446*