This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Munster. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary ad transparent.

Preface

Chapter 1 Covering Spaces

 1. The Definition of Riemann Surfaces

 2. Elementary Properties of Holomorphic Mappings

 3. Homotopy of Curves. The Fundamental Group

 4. Branched and Unbranched Coverings

 5. The Universal Covering and Covering Transformations

 6. Sheaves

 7. Analytic Continuation

 8. Algebraic Functions

 9. Differential Forms

 10. The Integration of Differential Forms

 11. Linear Differential Equations

Chapter 2 Compact Riemann Surfaces

 12. Cohomology Groups

 13. Dolbeault''s Lemma

 14. A Finiteness Theorem

 15. The Exact Cohomology Sequence

 16. The Riemann-Roch Theorem

 17. The Serre Duality Theorem

 18. Functions and Differential Forms with Prescribed Principal Parts

 19. Harmonic Differential Forms

 20. Abel''s Theorem

 21. The Jacobi Inversion Problem

Chapter 3 Non-compact Riemann Surfaces

 22. The Dirichlet Boundary Value Problem

 23. Countable Topology

 24. Weyl's Lemma

 25. The Runge Approximation Theorem

 26. The Theorems of Mittag-Leffler and Weierstrass

 27. The Riemann Mapping Theorem

 28. Functions with Prescribed Summands of Automorphy

 29. Line and Vector Bundles

 30. The Triviality of Vector Bundles

 31. The Riemann-Hilbert Problem

Appendix

 A. Partitions of Unity

 B. Topological Vector Spaces

References

Symbol Index

Author and Subject Index