**Mathematical Surveys and Monographs**

Volume: 76;
2000;
372 pp;
Hardcover

MSC: Primary 30; 32;

Print ISBN: 978-0-8218-1983-8

Product Code: SURV/76

List Price: $109.00

Individual Member Price: $87.20

**Electronic ISBN: 978-1-4704-1303-3
Product Code: SURV/76.E**

List Price: $109.00

Individual Member Price: $87.20

# Quasiconformal Teichmüller Theory

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*Frederick P. Gardiner; Nikola Lakic*

The Teichmüller space \(T(X)\) is the space of marked conformal structures on a given quasiconformal surface \(X\). This volume uses quasiconformal mapping to give a unified and up-to-date treatment of \(T(X)\). Emphasis is placed on parts of the theory applicable to noncompact surfaces and to surfaces possibly of infinite analytic type.

The book provides a treatment of deformations of complex structures on infinite Riemann surfaces and gives background for further research in many areas. These include applications to fractal geometry, to three-dimensional manifolds through its relationship to Kleinian groups, and to one-dimensional dynamics through its relationship to quasisymmetric mappings. Many research problems in the application of function theory to geometry and dynamics are suggested.

#### Table of Contents

# Table of Contents

## Quasiconformal Teichmuller Theory

- CONTENTS vii10 free
- PREFACE xiii16 free
- ACKNOWLEDGEMENTS xix22 free
- 1. QUASICONFORMAL MAPPING 124 free
- 2. RIEMANN SURFACES 1740
- 2.1 Conformal Structure 1841
- 2.2 Examples and Uniformization 1841
- 2.3 Extremal Length 2144
- 2.4 Teichmüller Space 2447
- 2.5 Metrics of Constant Curvature 2649
- 2.6 Thrice-Punctured Spheres 3053
- 2.7 Fuchsian Groups 3154
- 2.8 Types of Elements of PSX(2,R) 3760
- 2.9 Fundamental Domains 3861
- 2.10 Dimension of Quadratic Differentials 4164

- 3. QUADRATIC DIFFERENTIALS, PART I 4366
- 3.1 Integrable Quadratic Differentials 4669
- 3.2 Poincaré Theta Series 4972
- 3.3 Predual Space 5275
- 3.4 Closed Sets 5477
- 3.5 The Teichmuller Infinitesimal Norm 5881
- 3.6 Cross-Ratio Norm on Z(∞) 5982
- 3.7 Approximation by Rational Functions 6285
- 3.8 Rational Quadratic Differentials 6689
- 3.9 The Equivalence Theorem 6790
- 3.10 Vanishing Elements of Z(∞) 7093
- Appendix, Proof of the Equivalence Theorem 7396

- 4. QUADRATIC DIFFERENTIALS, PART II 83106
- 4.1 Horizontal Trajectories 84107
- 4.2 Geodesic Trajectories 86109
- 4.3 The Minimal Norm Property 89112
- 4.4 The Reich-Strebel Inequality 93116
- 4.5 Surfaces of Infinite Analytic Type 94117
- 4.6 The Main Inequality and Uniqueness 95118
- 4.7 The Frame Mapping Theorem 96119
- 4.8 Infinitesimal Frame Mapping 99122
- 4.9 The Fundamental Inequalities 101124
- 4.10 Teichmüller Contraction 102125
- 4.11 Strebel Points 104127
- 4.12 Teichmüller's Infinitesimal Metric 106129

- 5. TEICHMÜLLER EQUIVALENCE 109132
- 6. THE BERS EMBEDDING 125148
- 7. KOBAYASHI'S METRIC ON TEICHMÜLLER SPACE 145168
- 8. ISOMORPHISMS AND AUTOMORPHISMS 155178
- 8.1 Global to Local 155178
- 8.2 Automorphisms of Teichmüller Discs 157180
- 8.3 Rotational Transitivity 159182
- 8.4 Adjointness Theorem 161184
- 8.5 Isometries of Teichmüller Spaces 162185
- 8.6 The Isometry Property 163186
- 8.7 Nonsmoothness of the Norm 164187
- 8.8 Isometry Theorem for Genus Zero 166189
- 8.9 Riemann Surfaces of Finite Genus 170193

- 9. TEICHMÜLLER UNIQUENESS 177200
- 10. THE MAPPING CLASS GROUP 195218
- 11. JENKINS-STREBEL DIFFERENTIALS 207230
- 12. MEASURED FOLIATIONS 223246
- 13. OBSTACLE PROBLEMS 241264
- 13.1 Extremal Problem for the Disc 242265
- 13.2 Extremal Problem for a Surface 244267
- 13.3 Smoothing the Contours 245268
- 13.4 Boundedness of the Norm 245268
- 13.5 Schiffer and Beltrami Variations 248271
- 13.6 Existence 250273
- 13.7 Uniqueness 251274
- 13.8 Slit Mappings 253276
- 13.9 Trajectories around the Obstacle 254277

- 14. ASYMPTOTIC TEICHMULLER SPACE 257280
- 14.1 The Infinitesimal Theory 258281
- 14.2 Harmonic Beltrami Differentials 260283
- 14.3 The Earle- Nag Reflection 263286
- 14.4 Generalized Ahlfors-Weill Sections 266289
- 14.5 Bers' L-Operators 268291
- 14.6 Inverse Operators 269292
- 14.7 Manifold Structure 271294
- 14.8 Inequalities for Boundary Dilatation 275298
- 14.9 Contraction 276299
- 14.10 Extremality in AT 281304
- 14.11 Teichmüller's Metric 282305

- 15. ASYMPTOTICALLY EXTREMAL MAPS 285308
- 16. UNIVERSAL TEICHMÜLLER SPACE 299322
- 17. SUBSTANTIAL BOUNDARY POINTS 323346
- 18. EARTHQUAKE MAPPINGS 337360
- BIBLIOGRAPHY 357380
- INDEX 369392

#### Readership

Graduate students, research and applied mathematicians and physicists interested in functions of a complex variable, several complex variables and analytic spaces, particularly mathematical foundations of deformation theory.

#### Reviews

The goal of this book is stated in the excellent preface: ‘to provide background for applications of Teichmüller theory to dynamical systems’ … The extensive bibliography is instructive … a very interesting book … The treatment is clear and methodical … probably be possible to read this as an introductory text, yet there is much that is relatively new, innovative, and perhaps, percipient.

-- Bulletin of the LMS

[The authors] have produced a formidable treatise on the modern theories of quasiconformal mappings, Riemann surfaces and Teichmüller spaces. They have gathered, into a unified exposition, results which … have not previously been found in book form … Many of the approaches and results are new, others are more detailed than can be found elsewhere … this monograph is now the standard reference on two-dimensional quasiconformal mappings and Teichmüller theory and is likely to remain so for many years.

-- Mathematical Reviews

Brings to the literature the current state of the analytic theory of Teichmüuller spaces … a thorough report on the latest developments … a solid exposition of most of the classical foundations … [this book] is a real service to the community … an important addition to the literature … topics are discussed very thoroughly; some shed a lot of new light on the material.

-- Bulletin of the AMS