Author | Farkas, Hershel M. author |
---|---|
Title | Riemann Surfaces [electronic resource] / by Hershel M. Farkas, Irwin Kra |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1992 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2034-3 |
Descript | XVI, 366 p. online resource |
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 TheoryโA Quick Review -- II.2. Weylโs 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. Abelโs 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. Torelliโs Theorem -- IV Uniformization -- IV. 1. More on Harmonic Functions (A Quick Review) -- IV.2. Subharmonic Functions and Perronโs 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-RochโAn Alternate Approach -- IV. 11. Algebraic Function Fields in One Variable -- V Automorphisms of Compact SurfacesโElementary Theory -- V.l. Hurwitzโs Theorem -- V.2. Representations of the Automorphism Group on Spaces of Differentials -- V.3. Representation of 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