Functionals of Finite Riemann Surfaces

by Menahem Schiffer, Donald C. Spencer

Overview

This advanced monograph on finite Riemann surfaces, based on the authors' 1949–50 lectures at Princeton University, remains a fundamental book for graduate students. The Bulletin of the American Mathematical Society hailed the self-contained treatment as the source of "a plethora of ideas, each interesting in its own right," noting that "the patient reader will be richly rewarded."
Suitable for graduate-level courses, the text begins with three chapters that offer a development of the classical theory along historical lines, examining geometrical and physical considerations, existence theorems for finite Riemann surfaces, and relations between differentials. Subsequent chapters explore bilinear differentials, surfaces imbedded in a given surface, integral operators, and variations of surfaces and of their functionals. The book concludes with a look at applications of the variational method and remarks on generalization to higher dimensional Kahler manifolds.

Product Details

ISBN-13:	9780486795430
Publisher:	Dover Publications
Publication date:	06/01/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	464
File size:	37 MB
Note:	This product may take a few minutes to download.

About the Author

Menahem Max Schiffer (1911–97) taught at the Hebrew University of Jerusalem, Harvard, and Princeton before joining the faculty at Stanford University, where he was Chairman of the Mathematics Department from 1954 to 1959 and the Robert Grimmett Professor of Mathematics. His previous Dover title is Kernel Functions and Elliptic Differential Equations in Mathematics and Physics.
Donald Clayton Spencer (1921–2001) ranks among the most prominent American mathematicians of his generation. He taught at Princeton and Stanford and collaborated with Kunihiko Kodaira on the modern theory of deformation of complex structures. He is co-author of Dover's Advanced Calculus.

Table of Contents

Chapter 1 Geometrical and physical considerations 1

1.1 Conformal flatness. Beltrami's equation 1

1.2 Exterior differential forms 11

1.3 Differential forms on Riemann surfaces 15

1.4 Elementary topology of surfaces 17

1.5 Integration formulas 20

Chapter 2 Existence theorems for finite Riemann surfaces 25

2.1 Definition of a Riemann surface 25

2.2 The double of a finite Riemann surface 29

2.3 Hilbert space 33

2.4 Orthogonal projection 40

2.5 The fundamental lemma 42

2.6 The existence of harmonic differentials with prescribed periods 44

2.7 Existence of single-valued harmonic functions with singularities 48

2.8 Boundary-value problems by the method of orthogonal projection 51

2.9 Harmonic functions of a finite surface 58

2.10 The Uniformization Principle for finite surfaces 59

2.11 Conformal mapping onto canonical domains of higher genus 62

Chapter 3 Relations between differentials 64

3.1 Abelian differentials 64

3.2 The period matrix 71

3.3 Normalized differentials 72

3.4 Period relations 74

3.5 The order of a differential 76

3.6 The Riemann-Roch theorem for finite Riemann surfaces 78

3.7 Conformal mappings of a finite Riemann surface onto itself 83

3.8 Reciprocal and quadratic differentials 85

Chapter 4 Bilinear differentials 88

4.1 Bilinear differentials and reproducing kernels. 88

4.2 Definition of the Green's and Neumann's functions 93

4.3 Differentials of the first kind defined in terms of the Green's function 101

4.4 Differentials of the first kind defined in terms of the Neumann's function 105

4.5 Period matrices 107

4.6 Relations between the Green's and Neumann's functions 109

4.7 Canonical mapping functions 110

4.8 Classes of differentials 114

4.9 The bilinear differentials for the class F 117

4.10 Construction of the bilinear differential for the class M in terms of the Green's function 121

4.11 Construction of the bilinear differential for the class F 126

4.12 Properties of the bilinear differentials 129

4.13 Approximation of differentials 137

4.14 A special complete orthonormal system 138

Chapter 5 Surfaces imbedded in a given surface 143

5.1 One surface imbedded in another 143

5.2 Several surfaces imbedded in a given surface. 147

5.3 Fundamental identities 148

5.4 Inequalities for quadratic and Hermitian forms 153

5.5 Extension of a local complex analytic imbedding of one surface in another 158

5.6 Applications to schlicht functions 168

5.7 Extremal mappings 173

5.8 Non-schlicht mappings 178

Chapter 6 Integral operators 181

6.1 Definition of the operators T, T and S 181

6.2 Scalar products of transforms 185

6.3 The iterated operators 189

6.4 Spaces of piecewise analytic differentials 198

6.5 Conditions for the vanishing of a differential 199

6.6 Bounds for the operators T and T 208

6.7 Spectral theory of the t-operator 212

6.8 Spectral theory of the t-operator 219

6.9 Spectral theory of the s-operator 223

6.10 Minimum-maximum properties of the eigen-differentials 230

6.11 The Hilbert space with Dirichlet metric 233

6.12 Comparison with classical potential theory 241

6.13 Relation between the eigen-differentials of M and R ? M 244

6.14 Extension to disconnected surfaces 252

6.15 Representation of domain functionals of M in terms of the domain functionals of R 254

6.16 The combination theorem 262

Chapter 7 Variations of surfaces and of their functionals 273

7.1 Boundary variations 273

7.2 Variation of functionals as first terms of series developments 277

7.3 Variation by cutting a hole 283

7.4 Variation bv cutting a hole in a closed surface 290

7.5 Attaching a handle to a closed surface 293

7.6 Attaching a handle to a surface with boundary 299

7.7 Attaching a cross-capin 303

7.8 Interior deformation by attaching a cell. First method 310

7.9 Interior deformation by attaching a cell. Second method 314

7.10 The variation kernel 316

7.11 Identities satisfied by the variation kernel 323

7.12 Conditions for conformal equivalence under a deformation 331

7.13 Construction of the variation which preserves conformal type 334

7.14 Variational formulas for conformal mapping 347

7.15 Variations of boundary type 354

Chapter 8 Applications of the variational method 357

8.1 Identities for functionals 357

8.2 Hie coefficient problem for schlicht functions 364

8.3 Imbedding a circle in a given surface 376

8.4 Canonical cross-cuts on a surface R 385

8.5 Extremum problems in the conformal mapping of plane domains 396

Chapter 9 Remarks on generalization to higher dimensional Kähler manifolds 408

9.1 Kahler manifolds 408

9.2 Complex operators 415

9.3 Finite manifolds 420xs

9.4 Currents 423

9.5 Hermitian metrics 425

9.6 Dirichlet's principle for the real operators 425

9.7 Bounded manifolds 433

9.8 Dirichlet's principle for the complex operators 438

9.9 Bounded Kahler manifolds 439

9.10 Existence theorems on compact Kähler manifolds 440

9.11 The 1,-kernels on finite Kahler manifolds 443

9.12 Intrinsic definition of the operators 445

Index 448