Riemann Surfaces

by Farkas, Hershel M.

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Title Statements

Full title :Riemann Surfaces / by Hershel M. Farkas, Irwin Kra.

Summary

This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case. Basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the Abelian varities associated with these surfaces. Topics covered include existence of meromorphic functions, the Riemann -Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem. A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented. Alternate proofs for the most important results are included, showing the diversity of approaches to the subject. For this new edition, the material has been brought up- to-date, and erros have been corrected. The book should be of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics.

Edition Statements

Publication Date:1992.
Edition:Second edition

Publication Statements

Publisher(s):Springer New York
Place of Publication:New York, NY
Language:eng
Publication Date:1992.

The Physical Object

Description:1 online resource (xvi, 366 pages)

Edition Identifiers

ISBN:9781461220343, 1461220343
DOI:10.1007/978-1-4612-2034-3
oclc:ocn853270503
LC Classification:QA299.6-433
Dewey Decimal:515

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 Theory -- A Quick Review -- II. 2. Weyl'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 Riemann-Roch Theorem -- III. 5. Applications of the Riemann-Roch Theorem -- III. 6. Abel'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 -- III. 12. Torelli's Theorem -- IV Uniformization -- IV. 1. More on Harmonic Functions (A Quick Review) -- IV. 2. …

Subjects

Analyse globale (Mathématiques) Géométrie algébrique Geometry, Algebraic Global analysis (Mathematics) Groupes topologiques mathematics Mathématiques Topological groups