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AuthorFarkas, Hershel M. author
TitleRiemann Surfaces [electronic resource] / by Hershel M. Farkas, Irwin Kra
ImprintNew York, NY : Springer New York : Imprint: Springer, 1980
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Descript XI, 340 p. online resource


The present volume is the culmination often years' work separately and jointยญ ly. The idea of writing this book began with a set of notes for a course given by one of the authors in 1970-1971 at the Hebrew University. The notes were refined serveral times and used as the basic content of courses given subยญ sequently by each of the authors at the State University of New York at Stony Brook and the Hebrew University. In this book we present the theory of Riemann surfaces and its many difยญ ferent facets. We begin from the most elementary aspects and try to bring the reader up to the frontier of present-day research. We treat both open and closed surfaces in this book, but our main emphasis is on the compact case. In fact, Chapters III, V, VI, and VII deal exclusively with compact surfaces. Chapters I and II are preparatory, and Chapter IV deals with uniformization. All works on Riemann surfaces go back to the fundamental results of Rieยญ mann, Jacobi, Abel, Weierstrass, etc. Our book is no exception. In addition to our debt to these mathematicians of a previous era, the present work has been influenced by many contemporary mathematicians


0 An Overview -- 0.1 Topological Aspects, Uniformization, and Fuchsian Groups -- 0.2 Algebraic Functions -- 0.3. Abelian Varieties -- 0.4. More Analytic Aspects -- I Riemann Surfaces -- I.1. Definitions and Examples -- I.2. Topology of Riemann Surfaces -- I.3. Differential Forms -- I.4. Integration Formulae -- II Existence Theorems -- II.1. Hilbert Space Theoryโ{128}{148}A Quick Review -- II.2. Weylโ{128}{153}s Lemma -- II.3. The Hilbert Space of Square Integrable Forms -- II.4. Harmonic Differentials -- II.5. Meromorphic Functions and Differentials -- III Compact Riemann Surfaces -- III.1. Intersection Theory on Compact Surfaces -- III.2. Harmonic and Analytic Differentials on Compact Surfaces -- III.3. Bilinear Relations -- III.4. Divisors and the Riemannโ{128}{148}Roch Theorem -- III.5. Applications of the Riemannโ{128}{148}Roch Theorem -- III.6. Abelโ{128}{153}s Theorem and the Jacobi Inversion Problem -- III.7. Hyperelliptic Riemann Surfaces -- III.8. Special Divisors on Compact Surfaces -- III.9. Multivalued Functions -- III.10. Projective Imbeddings -- III.11. More on the Jacobian Variety -- IV Uniformization -- IV.1. More on Harmonic Functions (A Quick Review) -- IV.2. Subharmonic Functions and Perronโ{128}{153}s Method -- IV.3. A Classification of Riemann Surfaces -- IV.4. The Uniformization Theorem for Simply Connected Surfaces -- IV.5. Uniformization of Arbitrary Riemann Surfaces -- IV.6. The Exceptional Riemann Surfaces -- IV.7. Two Problems on Moduli -- IV.8. Riemannian Metrics -- IV.9. Discontinuous Groups and Branched Coverings -- IV.10. Riemannโ{128}{147}Rochโ{128}{148}An Alternate Approach -- IV.11. Algebraic Function Fields in One Variable -- V Automorphisms of Compact Surfaces Elementary Theory -- V.1. Hurwitzโ{128}{153}s Theorem -- V.2. Representations of the Automorphism Group on Spaces of Differentials -- V.3. Representations of Aut M on H>1(M) -- V.4. The Exceptional Riemann Surfaces -- VI Theta Functions -- VI.1. The Riemann Theta Function -- VI.2. The Theta Functions Associated with a Riemann Surface -- VI.3. The Theta Divisor -- VII Examples -- VII.1. Hyperelliptic Surfaces (Once Again) -- VII.2. Relations among Quadratic Differentials -- VII.3. Examples of Non-hyperelliptic Surfaces -- VII.4. Branch Points of Hyperelliptic Surfaces as Holomorphic Functions of the Periods -- VII.5. Examples of Prym Differentials

Mathematics Algebraic geometry Mathematical analysis Analysis (Mathematics) Mathematics Analysis Algebraic Geometry


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