Compact Riemann Surfaces

1. Front Matter

   Pages I-XVIII

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2. Topological Foundations

   Pages 1-16

3. Differential Geometry of Riemann Surfaces

   Pages 17-78

4. Harmonic Maps

   Pages 79-159

5. Teichmüller Spaces

   Pages 161-186

6. Geometric Structures on Riemann Surfaces

   Pages 187-265

7. Erratum to: Characterizing Programming Systems Allowing Program Self-reference

   • John Case, Samuel E. Moelius

   Pages 266-266

8. Back Matter

   Pages 267-281

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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. The analytic approach is likewise new as it is based on the theory of harmonic maps. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.

• Riemann surfaces
• Riemann-Roch theorem
• Teichmüller theory
• differential geometry
• harmonic maps
• hyberbolic geometry
• hyperbolic geometry

From the reviews:

"Compact Riemann Surfaces: An Introduction to Contemporary Mathematics starts off with a wonderful Preface containing a good deal of history, as well as Jost’s explicit dictum that there are three foci around which the whole subject revolves … . Jost’s presentation is quite accessible, modulo a lot of diligence on the part of the reader. It’s a very good and useful book, very well-written and thorough." (Michael Berg, MathDL, April, 2007)

From the reviews of the third edition:

“Geometrical facts about Riemann surfaces are as ‘nice’ as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem … is a prime example of this influence. This book is amazing, very well written, accessible and works as a first course on Riemannian Surfaces I recommend to the all readers interested in Geometry and Riemannian Geometry.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)

• Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

  Jürgen Jost