Geometric Function Theory

* Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research* Methodically designed with individual chapters containing a rich collection of exercises, examples, and illustrations

Preface

xi

Part I Classical Function Theory

Overview

3

(154)

1 Invariant Geometry

5

(24)

1.1 Conformality and Invariance

5

(4)

1.2 Bergman's Construction

9

(6)

1.3 Calculation of the Bergman Kernel for the Disk

15

(6)

1.3.1 Construction of the Bergman Kernel for the Disk by Conformal Invariance

16

(1)

1.3.2 Construction of the Bergman Kernel by means of an Orthonormal System

17

(1)

1.3.3 Construction of the Bergman Kernel by way of Differential Equations

18

(3)

1.4 A New Application

21

(3)

1.5 An Application to Mapping Theory

24

(2)

Problems for Study and Exploration

26

(3)

2 Variations on the Theme of the Schwarz Lemma

29

(36)

2.1 Introduction

29

(3)

2.2 Other Versions of Schwarz's Lemma

32

(1)

2.3 A Geometric View of the Schwarz Lemma

33

(17)

2.3.1 Geometric Ideas

33

(2)

2.3.2 Calculus in the Complex Domain

35

(2)

2.3.3 Isometries

37

(2)

2.3.4 The Poincaré Metric

39

(8)

2.3.5 The Schwarz Lemma

47

(3)

2.4 Ahlfors's Version of the Schwarz Lemma

50

(4)

2.5 Liouville and Picard Theorems

54

(4)

2.6 The Schwarz Lemma at the Boundary

58

(5)

Problems for Study and Exploration

63

(2)

3 Normal Families

65

(18)

3.1 Introduction

65

(1)

3.2 Topologies on the Space of Holomorphic Functions

66

(1)

3.3 Normal Families in Their Natural Context

67

(6)

3.4 Advanced Results on Normal Families

73

(5)

3.5 Robinson's Heuristic Principle

78

(2)

Problems for Study and Exploration

80

(3)

4 The Riemann Mapping Theorem and Its Generalizations

83

(26)

4.1 The Riemann Mapping Theorem by way of the Green's Function

84

(1)

4.2 Canonical Representations for Multiply Connected Regions

85

(1)

4.3 Review of Some Topological Ideas

86

(4)

4.3.1 Cycles and Periods

86

(2)

4.3.2 Harmonic Functions

88

(2)

4.4 Uniformization of Multiply Connected Domains

90

(4)

4.5 The Ahlfors Map

94

(10)

4.6 The Uniformization Theorem

104

(2)

Problems for Study and Exploration

106

(3)

5 Boundary Regularity of Conformal Maps

109

(26)

5.1 Continuity to the Boundary

110

(8)

5.2 Preliminary Facts about Boundary Smoothness

118

(9)

5.3 Smoothness of Conformal Mappings

127

(4)

Problems for Study and Exploration

131

(4)

6 The Boundary Behavior of Holomorphic Functions

135

(22)

6.0 Introductory Remarks

135

(1)

6.1 Review of the Classical Theory of Hp Spaces on the disk

136

(10)

6.2 The Lindelöf Principle

146

(6)

Problems for Study and Exploration

152

(5)

Part II Real and Harmonic Analysis

Overview

157

(48)

7 The Cauchy-Riemann Equations

159

(10)

7.1 Introduction

160

(1)

7.2 Solution of the Inhomogeneous Cauchy-Riemann Equations

161

(2)

7.3 Development and Application of the Equation

163

(3)

Problems for Study and Exploration

166

(3)

8 The Green's Function and the Poisson Kernel

169

(16)

8.1 The Laplacian and Its Fundamental Solution

169

(7)

8.2 The Green's Function and Consequences

176

(2)

8.3 Calculation of the Poisson Kernel

178

(4)

Problems for Study and Exploration

182

(3)

9 Harmonic Measure

185

(20)

9.1 Introductory Remarks

185

(3)

9.2 The Idea of Harmonic Measure

188

(2)

9.3 Some Examples

190

(4)

9.4 Hadamard's Three Circles Theorem

194

(3)

9.5 A Discussion of Interpolation of Linear Operators

197

(3)

9.6 The F. and M. Riesz Theorem

200

(3)

Problems for Study and Exploration

203

(2)

10 Conjugate Functions and the Hilbert Transform

205

(20)

10.0 Introduction

205

(1)

10.1 Discovering the Hilbert Transform

206

(1)

10.2 The Modified Hilbert Transform

207

(2)

10.3 The Hilbert Transform and Fourier Series

209

(10)

10.4 Proof of Theorem 10.3.8

219

(3)

Problems for Study and Exploration

222

(3)

11 Wolff's Proof of the Corona Theorem

225

(30)

11.1 Introductory Remarks

226

(2)

11.2 The Banach Algebra Hinfinity

228

(1)

11.3 Statement of the Corona Theorem

228

(1)

11.4 Carleson Measures

229

(5)

11.5 A Key Technical Lemma

234

(6)

11.6 Proof of the Corona Theorem

240

(5)

11.7 Proof of Lemma 11.6.1

245

(5)

Problems for Study and Exploration

250

(5)

Part III Algebraic Topics

Overview

255

(2)

12 Automorphism Groups of Domains in the Plane

257

(24)

12.1 Introductory Concepts

257

(2)

12.2 Noncompact Automorphism Groups

259

(6)

12.3 The Dimension of the Automorphism Group

265

(3)

12.4 The Iwasawa Decomposition

268

(3)

12.5 General Properties of Holomorphic Maps

271

(7)

Problems for Study and Exploration

278

(3)

13 Cousin Problems, Cohomology, and Sheaves

281

(22)

13.1 The Cousin Problems

282

(5)

13.2 A Few Words About Sheaves

287

(14)

Problems for Study and Exploration

301

(2)

Bibliography

303

(4)

Index

307

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Geometric Function Theory

Summary

Table of Contents

Supplemental Materials

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