This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Münster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one.

From the reviews: "This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces."--MATHEMATICAL REVIEWS

Inhalt

1 Covering Spaces.- The Definition of Riemann Surfaces.- Elementary Properties of Holomorphic Mappings.- Homotopy of Curves. The Fundamental Group.- Branched and Unbranched Coverings.- The Universal Covering and Covering Transformations.- Sheaves.- Analytic Continuation.- Algebraic Functions.- Differential Forms.- . The Integration of Differential Forms.- . Linear Differential Equations.- 2 Compact Riemann Surfaces.- . Cohomology Groups.- . Dolbeault's Lemma.- . A Finiteness Theorem.- . The Exact Cohomology Sequence.- . The Riemann-Roch Theorem.- . The Serre Duality Theorem.- . Functions and Differential Forms with Prescribed Principal Parts.- . Harmonic Differential Forms.- . Abel's Theorem.- . The Jacobi Inversion Problem.- 3 Non-compact Riemann Surfaces.- . The Dirichlet Boundary Value Problem.- . Countable Topology.- . Weyl's Lemma.- . The Runge Approximation Theorem.- . The Theorems of Mittag-Leffler and Weierstrass.- . The Riemann Mapping Theorem.- . Functions with Prescribed Summands of Automorphy.- . Line and Vector Bundles.- . The Triviality of Vector Bundles.- . The Riemann-Hilbert Problem.- A. Partitions of Unity.- B. Topological Vector Spaces.- References.- Symbol Index.- Author and Subject Index.