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...
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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.
Preface1 The Grotzch argument2 Geometric definition of quasiconformal maps3 Analytic properties of quasiconformal maps4 Quasi-isometries and quasisymmetric maps5 The Beltrami differential equation6 Holomorphic motions and applications7 Teichmuller spaces8 Extremal quasiconformal mappings9 Unique extremality10 Isomorphisms of Teichmuller space11 Local rigidity of Teichmuller spacesReferencesIndex1 The Grotzsch argument 11.1 Maps on rectangles 11.2 Some definitions 21.3 Solving the Grotzsch problem 31.4 Composed mappings 51.5 Riemann surfaces 72 Geometric definition of quasiconformal maps 102.1 Extremal length 112.2 Curve families 132.3 Geometric definition of quasiconformal maps 143 Analytic properties of quasiconformal maps 193.1 Analytic definition and corollaries 193.2 Extremal ring domains 233.3 Holder continuity 263.4 Compactness properties of quasiconformal maps 294 Quasi-isometries and quasisymmetric maps 314.1 Cross-ratio 314.2 Quasisymmetric maps 324.3 Quasi-isometry 364.4 The barycentric extension 415 The Beltrami differential equation 485.1 Integral transforms 485.2 Solution of the Beltrami equation 515.3 Dependence on Beltrami coefficients 576 Holomorphic motions and applications 646.1 Holomorphic motions 646.2 Equivariant extensions 686.3 Area distortion 717 Teichmuller spaces 767.1 Universal Teichmuller space 767.2 Teichmuller space of a Riemann surface 777.3 Teichmuller metric 807.4 The Teichmuller space of a torus 847.5 Schwarzian derivatives and quadratic differentials 887.6 The Bers embedding 927.7 Complex structure onTeichmuller space 978 Extremal quasiconformal mappings 1048.1 Examples of extremal mappings 1048.2 The Hamilton-Krushkal condition 1098.3 The Main Inequality 1168.4 Sufficiency of the Hamilton-Krushkal condition 1189 Unique extremality 1239.1 The frame mapping condition 1239.2 Some necessary conditions for unique extremality 1309.3 Delta inequalities 1329.4 Beltrami differentials with constant modulus 1359.5 Beltrami differentials with non-constant modulus 1399.6 Hahn-Banach extensions 14510 Isomorphisms of Teichmuller space 14910.1 The Kobayashi metric 14910.2 Equimeasurability 15210.3 Isometries of Bergman spaces 15610.4 Geometric isometries in the general case 15910.5 Biholomorphic maps between Teichmuller spaces 17111 Local rigidity of Teichmuller spaces 17311.1 Bergman kernels 17311.2 Operators on A1 (M) 17711.3 An isomorphism between A1 (M) and l1 18011.4 Local bi-Lipschitz equivalence of Teichmuller spaces 182References 184Index 188
