Springer 1997, 3. Auflage, 2006

Table of Contents

1 Topological Foundations

1.1 Manifolds and Differentiable Manifolds
1.2 Homotopy of Maps. The Fundamental Group
1.3 Coverings
1.4 Global Continuation of Functions on Simply-Connected Manifolds

2 Differential Geometry of Riemann Surfaces

2.1 The Concept of a Riemann Surface
2.2 Some Simple Properties of Riemann Surfaces
2.3 Metrics on Riemann Surfaces
    2.3.A Triangulations of Compact Riemann Surfaces
2.4 Discrete Groups of Hyperbolic Isometries. Fundamental Polygons. Some Basic Concepts of Surface Topology and Geometry
    2.4.A The Topological Classification of Compact Riemann Surfaces
2.5 The Theorems of Gauss-Bonnet and Riemann-Hurwitz
2.6 A General Schwarz Lemma
2.7 Conformal Structures on Tori

3 Harmonic Maps

3.1 Review: Banach and Hilbert Spaces. The Hilbert Space L²
3.2 The Sobolev Space W ^1,2 = H ^1,2
3.3 The Dirichlet Principle. Weak Solutions of the Poisson Equation
3.4 Harmonic and Subharmonic Functions
3.5 The C^α-Regularity Theory
3.6 Maps Between Surfaces. The Energy Integral. Definition and Simple Properties of Harmonic Maps
3.7 Existence of Harmonic Maps
3.8 Regularity of Harmonic Maps
3.9 Uniqueness of Harmonic Maps
3.10 Harmonic Diffeomorphisms
3.11 Metrics and Conformal Structures

4 Teichmüller Spaces

4.1 The Basic Definitions
4.2 Harmonie Maps, Conformal Structures and Holomorphic Quadratic Differentials. Teichmüller's Theorem
4.3 Fenchel-Nielsen Coordinates. An Alternative Approach to the Topology of Teichmüller Space
4.4 Uniformization of Compact Riemann Surfaces

5 Geometric Structures on Riemann Surfaces

5.1 Preliminaries: Cohomology and Homology Groups
5.2 Harmonic and Holomorphic Differential Forms on Riemann Surfaces
5.3 The Periods of Holomorphic and Meromorphic Differential Forms
5.4 Divisors. The Riemann-Roch Theorem
5.5 Holomorphic l-Forms and Metrics on Compact Riemann Surfaces
5.6 Divisors and Line Bundles
5.7 Projective Embeddings
5.8 Algebraic Curves
5.9 Abel's Theorem and the Jacobi Inversion Theorem
5.10 Elliptic Curves