Conformally Invariant Metrics and Quasiconformal Mappings

Description

This book is an introduction to the theory of quasiconformal and quasiregular mappings in the euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop this theory as the literature shows. The authors' approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role throughout the whole book. The intended readership consists of mathematicians from beginning graduate students to researchers. The prerequisite requirements are modest: only some familiarity with basic ideas of real and complex analysis is expected. Preface Notation and Terminology Contents List of Figures Part I Introduction and Review 1 Introduction 2 A Survey of Quasiregular Mappings Part II Conformal Geometry 3 Möbius Transformations 3.1 Poincaré Extension 3.1.1 Stereographic Projection 3.1.2 Balls in the Chordal Metric 3.1.3 Absolute Ratio 3.2 Automorphisms of the Unit Ball 3.2.1 The Lipschitz Constant of Ta|Bn 3.2.2 The Ahlfors Bracket 3.3 Chordal Isometries 4 Hyperbolic Geometry 4.1 The Poincaré Half Space 4.2 The Poincaré Unit Ball 4.3 Bounds for Hyperbolic Distance 5 Generalized Hyperbolic Geometries 5.1 The Quasihyperbolic Metric 5.2 Möbius Invariant Metrics 5.2.1 The Point-Pair Invariant mG 5.2.2 The Symmetric Ratio 5.2.3 The Generalized Hyperbolic Metric ρG 5.3 Properties of Generalized Hyperbolic Metrics 6 Metrics and Geometry 6.1 Uniform Domains and Generalizations 6.1.1 Ptolemy's Theorem 6.2 Whitney Squares and (a,b,s)-Admissible Families 6.2.1 Whitney Decomposition 6.2.2 (a,b,s)-Admissible Families 6.3 Harnack Functions Part III Modulus and Capacity 7 The Modulus of a Curve Family 7.1 Basic Properties of the Modulus 7.1.1 The Cylinder 7.1.2 The Spherical Ring 7.1.3 Constants and Dimension 7.2 Comparison Principle for the Modulus 7.2.1 The Modulus of a Ring 7.3 Grötzsch and Teichmüller Rings 7.4 Hypergeometric Functions and Elliptic Integrals 7.4.1 Elliptic Integrals and γ2(s) 7.4.2 The Gaussian Hypergeometric Function 7.4.3 Landen Transformation 7.4.4 Arithmetic-Geometric Mean 7.4.5 Modular Equations 7.4.6 Jacobi's Infinite Products 7.4.7 Ramanujan's Approximation of μ(r) [29, 5.51], [64, p. 91, (2.4)] 8 The Modulus as a Set Function 8.1 The Construction of c(E) 8.1.1 Proof of Theorem 8.1 8.2 Metric Concentration of Sets 8.2.1 Thickness and Capacity 8.2.2 Conformal Invariants in the Plane 8.2.3 Reduced Modulus 8.3 Tubular Neighborhoods 9 The Capacity of a Condenser 9.1 Spherical Symmetrization 9.2 Grötzsch and Teichmüller Condensers 9.3 Hyperbolic Metric and Capacity 9.4 Dimension Cancellation and Special Functions 10 Conformal Invariants 10.1 Two Conformal Metrics 10.2 Ferrand's Modulus Metric 10.3 Teichmüller's Function 10.4 QED Domains 10.5 Capacitary Geometry Part IV Intrinsic Geometry 11 Hyperbolic Type Metrics 11.1 Metrics Determined by One Boundary Point 11.2 Metrics Determined by Two Boundary Points 11.2.1 Special Values of sBn 11.2.2 Ptolemy-Alhazen Problem and s-Metric 12 Comparison of Metrics 12.1 The Unit Ball 12.2 The Upper Half Space 12.3 General Domains 13 Local Convexity of Balls 13.1 Distance Ratio Metric 13.2 Quasihyperbolic Balls 13.3 Apollonian Metric 13.4 Seittenranta Metric 14 Inclusion Results for Balls 14.1 The Punctured Space 14.2 The Upper Half-Space Part V Quasiregular Mappings 15 Basic Properties of Quasiregular Mappings 15.1 Topological Properties of Discrete Open Mappings 15.1.1 An Open Problem 15.2 Path Lifting 15.3 Analytic Properties of Quasiregular Mappings 15.4 Curve Families and Quasiconformal Mappings 15.4.1 Ferrand's Problem 15.4.2 Open Problem 16 Distortion Theory 16.1 The Schwarz Lemma and Quasiregular Maps 16.2 Bounds for Moduli of Continuity 16.3 The Schwarz Lemma in the Planar Case 16.3.1 Summary of Main Ideas 16.4 Further Results 16.4.1 An Open Problem 17 Dimension-Free Theory 17.1 Quasiregular Mappings and Harnack Functions 17.2 Quasihyperbolic Metric and Quasiregular Mapping 18 Metrics and Maps 18.1 An Open Problem on Quasicircles 19 Teichmüller's Displacement Problem 19.1 On Krzyż's Constant Part VI Solutions 20 Solutions to Exercises 20.1 Solution to Exercises in Part II 20.2 Solution to Exercises in Part III 20.3 Solution to Exercises in Part IV 20.4 Solution to Exercises in Part V A Some Open Problems Problems from [553, p. 193] New Problems B Formulary Hyperbolic Functions and Their Inverses Some Biographical Data References Index Author Index