Quasiconformal Maps and Teichmüller Theory »

Based on a series of graduate lectures given by Vladimir Markovic at the University of Warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and Teichmüller theory. Assuming some familiarity with Riemann surfaces and hyperbolic geometry, topics covered include the Grötzch argument, analytical properties of quasiconformal maps, the Beltrami differential equation, holomorphic motions and Teichmüller spaces. Where proofs are omitted, references to where they may be found are always given, and the text is clearly illustrated throughout with diagrams, examples, and exercises for the reader.

Preface

1 The Grotzch argument

2 Geometric definition of quasiconformal maps

3 Analytic properties of quasiconformal maps

4 Quasi-isometries and quasisymmetric maps

5 The Beltrami differential equation

6 Holomorphic motions and applications

7 Teichmuller spaces

8 Extremal quasiconformal mappings

9 Unique extremality

10 Isomorphisms of Teichmuller space

11 Local rigidity of Teichmuller spaces

References

Index

1 The Grotzsch argument 1

1.1 Maps on rectangles 1

1.2 Some definitions 2

1.3 Solving the Grotzsch problem 3

1.4 Composed mappings 5

1.5 Riemann surfaces 7

2 Geometric definition of quasiconformal maps 10

2.1 Extremal length 11

2.2 Curve families 13

2.3 Geometric definition of quasiconformal maps 14

3 Analytic properties of quasiconformal maps 19

3.1 Analytic definition and corollaries 19

3.2 Extremal ring domains 23

3.3 Holder continuity 26

3.4 Compactness properties of quasiconformal maps 29

4 Quasi-isometries and quasisymmetric maps 31

4.1 Cross-ratio 31

4.2 Quasisymmetric maps 32

4.3 Quasi-isometry 36

4.4 The barycentric extension 41

5 The Beltrami differential equation 48

5.1 Integral transforms 48

5.2 Solution of the Beltrami equation 51

5.3 Dependence on Beltrami coefficients 57

6 Holomorphic motions and applications 64

6.1 Holomorphic motions 64

6.2 Equivariant extensions 68

6.3 Area distortion 71

7 Teichmuller spaces 76

7.1 Universal Teichmuller space 76

7.2 Teichmuller space of a Riemann surface 77

7.3 Teichmuller metric 80

7.4 The Teichmuller space of a torus 84

7.5 Schwarzian derivatives and quadratic differentials 88

7.6 The Bers embedding 92

7.7 Complex structure onTeichmuller space 97

8 Extremal quasiconformal mappings 104

8.1 Examples of extremal mappings 104

8.2 The Hamilton-Krushkal condition 109

8.3 The Main Inequality 116

8.4 Sufficiency of the Hamilton-Krushkal condition 118

9 Unique extremality 123

9.1 The frame mapping condition 123

9.2 Some necessary conditions for unique extremality 130

9.3 Delta inequalities 132

9.4 Beltrami differentials with constant modulus 135

9.5 Beltrami differentials with non-constant modulus 139

9.6 Hahn-Banach extensions 145

10 Isomorphisms of Teichmuller space 149

10.1 The Kobayashi metric 149

10.2 Equimeasurability 152

10.3 Isometries of Bergman spaces 156

10.4 Geometric isometries in the general case 159

10.5 Biholomorphic maps between Teichmuller spaces 171

11 Local rigidity of Teichmuller spaces 173

11.1 Bergman kernels 173

11.2 Operators on A¹ (M) 177

11.3 An isomorphism between A¹ (M) and l¹ 180

11.4 Local bi-Lipschitz equivalence of Teichmuller spaces 182

References 184

Index 188