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
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