It is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generation of mathematicians from (newly) adjacent fields (for example, those interested in hyperbolic manifolds and iterations of rational maps) and young physicists who have been convinced (certainly not by mathematicians) that compact Riemann surfaces may play an important role in their (string) universe. We hope that non-mathematicians as well as mathematicians (working in nearby areas to the central topic of this book) will also learn part of this subject for the sheer beauty and elegance of the material (work of Weierstrass, Jacobi, Riemann, Hilbert, Weyl) and as healthy exposure to the way (some) mathematicians write about mathematics.
本书为英文版。
Preface to the Second Edition
Preface to the First Edition
Commonly Used Symbols
CHAPTER 0
An Overview
0.1. Topological Aspects, Uniformization, and Fuchsian Groups
0.2. Algebraic Functions
0.3. Abelian Varieties
0.4. More Analytic Aspects
CHAPTER I
Riemann Surfaces
1.1. Definitions and Examples
I.2. Topology of Riemann Surfaces
I.3. Differential Forms
I.4. Integration Formulae
CHAPTER II
Existence Theorems
II.1. Hilbert Space Theory--A Quick Review
II.2. Weyrs Lemma
I1.3. The Hilbert Space of Square lntegrable Forms
II.4. Harmonic Differentials
II.5. Meromorphic Functions and Differentials
CHAPTER III
Compact Riemann Surfaces
III.1. Intersection Theory on Compact Surfaces
III.2. Harmonic and Analytic Differentials on Compact Surfaces
III.3. Bilinear Relations
III.4. Divisors and the Riemann-Roch Theorem
III.5. Applications of the Riemann-Roch Theorem
III.6. Abel's Theorem and the Jacobi Inversion Problem
III.7. Hyperelliptic Riemann Surfaces
III.8. Special Divisors on Compact Surfaces
III.9. Multivalued Functions
III.10. Projective Imbeddings
IIl.11. More on the Jacobian Variety
III.12. Torelli's Theorem
CHAPTER IV
Uniformization
IV.I. More on Harmonic Functions (A Quick Review)
IV.2. Subharmonic Functions and Perron's Method
IV.3. A Classification of Riemann Surfaces
IV.4. The Uniformization Theorem for Simply Connected Surfaces
IV.5. Uniforrnization of Arbitrary Riemann Surfaces
IV.6. The Exceptional Riemann Surfaces
IV.7. Two Problems on Moduli
IV.8. Riemannian Metrics
IV.9. Discontinuous Groups and Branched Coverings
IV.10. Riemann-Roch--An Alternate Approach
IV.11. Algebraic Function Fields in One Variable
CHAPTER V
Automorphisms of Compact Surfaces--Elementary Theory
V.l. Hurwitz's Theorem
V.2. Representations of the Automorphism Group on Spaces of Differentials
V.3. Representation of Aut M on Ht(M)
V.4. The Exceptional Riemann Surfaces
CHAPTER VI
Theta Functions
VI.1. The Riemann Theta Function
VI.2. The Theta Functions Associated with a Riemann Surface
VI.3. The Theta Divisor
CHAPTER VII
Examples
VII.1. Hyperelliptic Surfaces (Once Again)
VII.2. Relations Among Quadratic Differentials
VII.3. Examples of Non-hyperelliptic Surfaces
VII.4. Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods
VII.5. Examples of Prym Differentials
VII.6. The Trisecant Formula
Bibliography
Index
