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Unique among textbooks on the topic: includes an introduction to Teichmüller theory
The new edition is expanded and rewritten to improve the presentation
Systematically explores the connection between Riemann surfaces and other fields of mathematics
Can serve as an introduction to contemporary mathematics as a whole, as it touches on a broad variety of topics
Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of mathematics. It can serve as an introduction to contemporary mathematics as a whole as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. It is unique among textbooks on Riemann surfaces in including an introduction to Teichmüller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.
Preface
1. Topological Foundations
1.1 Manifolds and differential 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 A Triangulations of compact Riemann surfaces
2.4 Discrete groups of hyperbolic isometries. Fundamental polygons. Some basic concepts of surface topology and geometry
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 W1,2=H1,2
3.3 The Dirichlet principle. Weak solutions of the Poisson equation
3.4 Harmonic and subharmonic functions
3.5 The Ca-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 Harmonic 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 diferential forms
5.4 Divisors. The Riemann-Roch theorem
5.5 Holomorphic 1-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 inversiontheorem
5.10 Elliptic curves
Sources and references
Bibliography
Index of notation
Index