The Theory and Practice of Conformal Geometry

by Steven G. Krantz

Overview

In this original text, prolific mathematics author Steven G. Krantz addresses conformal geometry, a subject that has occupied him for four decades and for which he helped to develop some of the modern theory. This book takes readers with a basic grounding in complex variable theory to the forefront of some of the current approaches to the topic. "Along the way," the author notes in his Preface, "the reader will be exposed to some beautiful function theory and also some of the rudiments of geometry and analysis that make this subject so vibrant and lively."
More up-to-date and accessible to advanced undergraduates than most of the other books available in this specific field, the treatment discusses the history of this active and popular branch of mathematics as well as recent developments. Topics include the Riemann mapping theorem, invariant metrics, normal families, automorphism groups, the Schwarz lemma, harmonic measure, extremal length, analytic capacity, and invariant geometry. A helpful Bibliography and Index complete the text.

Product Details

ISBN-13:	9780486793443
Publisher:	Dover Publications
Publication date:	02/17/2016
Series:	Aurora: Dover Modern Math Originals Series
Pages:	304
Product dimensions:	5.90(w) x 8.90(h) x 0.70(d)

About the Author

Steven G. Krantz is Professor of Mathematics at Washington University in St. Louis. He is Editor of the Notices of the American Mathematical Society and the author of many books in mathematics, including Real Analysis and Foundations: Third Edition, The Proof is in the Pudding: The Changing Nature of Mathematical Proof, and A Mathematician Comes of Age.

Table of Contents

Preface xi

1 The Riemann Mapping Theorem 1

1.0 Introduction 2

1.1 The Proof of the Analytic Form of the Riemann Mapping Theorem 2

Appendix: Traditional Proof of the Riemann Mapping Theorem 4

1.2 A New Proof of the RMT 7

1.2.1 Some Definitions 8

1.2.2 A Sketch of Thurston's Idea 11

1.2.3 Details of the Proof 11

1.2.4 Convergence of Circle Packings to the Riemann Mapping 13

1.2.5 The Main Result 14

1.3 The Riemann Mapping Theorem by Way of the Green's Function 16

1.4 The Ahlfors Map 18

Appendix to Section 1.4 28

1.5 Canonical Representations for Multiply Connected Domains 29

1.6 Some Basic Topological Ideas 31

1.6.1 Cycles and Periods 31

1.6.2 Harmonic Functions 34

1.7 Uniformization of Multiply Connected Domains 36

1.8 The Uniformization Theorem 42

2 Invariant Metrics 47

2.1 The Carathéodory Metric 48

2.2 The Kobayashi Metric 50

2.3 Completeness of the Carathéodory and Kobayashi Metrics 58

3 Normal Families 77

3.1 Montel's Theorem 78

3.2 Another Look at Normal Families 83

3.3 Normal Families in Their Natural Context 85

3.4 Advanced Results on Normal Families 92

3.5 Robinson's Heuristic Principle 97

4 Automorphism Groups 101

4.1 Introductory Concepts 102

4.2 Noncompact Automorphism Groups 104

4.3 The Dimension of the Automorphism Group 112

4.4 The Iwasawa Decomposition 115

4.5 General Properties of Holomorphic Maps 120

5 The Schwarz Lemma 129

5.1 Introduction to Schwarz 130

5.2 The Geometry of the Schwarz Lemma 133

5.2.1 Metrics 133

5.2.2 Calculus from the Complex Viewpoint 136

5.2.3 Isometries 138

5.2.4 The Poincaré Metric 140

5.2.5 The Spirit of the Schwarz Lemma 149

5.3 The Schwarz Lemma According to Ahlfors 152

Appendix: A Curvature Calculation 157

An Intrinsic Look at Curvature 157

Curvature on Planar Domains 164

5.4 Another Look at Schwarz's Lemma 166

6 Harmonic Measure 169

6.1 The Idea, of Harmonic Measure 170

6.2 Some Examples 172

6.3 Hadamard's Three-Circles Theorem 177

6.4 A Discussion of Interpolation of Linear Operators 182

6.5 The F. and M. Riesz Theorem 185

7 Extremal Length 191

7.1 Some Definitions 192

7.2 The Conformal Invariance of Extremal Length 193

7.3 Some Examples 193

8 Analytic Capacity 197

8.1 Calculating Analytic Capacity 199

8.2 Analytic Capacity and Removability 201

9 Invariant Geometry 213

9.1 Conformality and Invariance 214

9.2 Bergman's Construction 218

9.3 Calculation of the Bergman Kernel for the Disc 225

9.3.1 Construction of the Bergman Kernel for the Disc by Conformal Invariance 226

9.3.2 Construction of the Bergman Kernel by Means of an Orthonormal System 227

9.3.3 Construction of the Bergman Kernel by way of Differential Equations 229

9.4 A New Application 232

9.5 An Application to Mapping Theory 237

10 A New Look at the Schwarz Lemma 241

10.1 The Boundary Schwarz Lemma 241

10.2 Liouville's and Picard's Theorems 247

10.3 Harmonic Functions 252

10.4 Another Look at the Boundary Schwarz Lemma 255

10.4.1 Hopf's Lemma 255

10.5 Ideas of Löwner and Veiling 256

10.6 The Schwarz Lemma on the Boundary Redux 257

10.7 Chelst's Point of View 258

10.8 Several Complex Variables 261

Bibliography 267

Index 275