Overview

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.

Full Product Details

Author:   Hershel M. Farkas ,  Irwin Kra
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   2nd ed. 1992
Volume:   71
Dimensions:   Width: 15.50cm , Height: 2.20cm , Length: 23.50cm
Weight:   1.590kg
ISBN:  

9780387977034

ISBN 10:   0387977031
Pages:   366
Publication Date:   23 December 1991
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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. Subharmonic Functions and Perron’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. Riemann-Roch—An Alternate Approach.- IV. 11. Algebraic Function Fields in One Variable.- V Automorphisms of Compact Surfaces—Elementary Theory.- V.l. Hurwitz’s Theorem.- V.2. Representations of the Automorphism Group on Spaces of Differentials.- V.3. Representationof Aut M on H1(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.- VII.6. The Trisecant Formula.

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