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内容简介

Uniformization of Compact Riemann Surfaces Geometric Structures on Riemann Surfaces、Preliminaries： Cohomology and Homology Groups、Harmonic and Holomorphic Differential Forms on Riemann Surfaces、The Periods of Holomorphic and Meromorphic Differential Forms、Divisors. The Riemann-Roch Theorem、Holomorphic 1-Forms and Metrics on Compact Riemann Surfaces、Divisors and Line Bundles等。

作者简介

作者：(德国)乔斯特(Jost.J.)

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目录

Preface
1 Topological Foundations
1.1 Manifolds and Differentiable Manifolds
1.2 Homotopy of Maps. The Fundamental Group
1.3 Coverings
1.4 Global Continuation of Functions on Simply-Connected Manifolds
2 Differential Geometry of Riemann Surfaces
2.1 The Concept of a Riemann Surface
2.2 Some Simple Properties of Riemann Surfaces
2.3 Metrics on Riemann Surfaces
2.3 A Triangulations of Compact Riemann Surfaces
2.4 Discrete Groups of Hyperbolic Isometries. Fundamental Polygons. Some Basic Concepts of Surface Topology and Geometry.
2.4 A The Topological Classification of Compact Riemann Surfaces
2.5 The Theorems of Gauss-Bonnet and Riemann-Hurwitz
2.6 A General Schwarz Lemma
2.7 Conformal Structures on Tori
3 Harmonic Maps
3.1 Review: Banach and Hilbert Spaces. The Hilbert Space L2
3.2 The Sobolev Space W1 2=H1 2
3.3 The Dirichlet Principle. Weak Solutions of the Poisson Equation
3.4 Harmonic and Subharmonic Functions
3.5 The Ca Regularity Theory
3.6 Maps Between Surfaces. The Energy Integral. Definition and Simple Properties of Harmonic Maps
3.7 Existence of Harmonic Maps
3.8 Regularity of Harmonic Maps
3.9 Uniqueness of Harmonic Maps
3.10 Harmonic Diffeomorphisms
3.11 Metrics and Conformal Structures
4 Teichmuller Spaces
4.1 The Basic Definitions
4.2 Harmonic Maps, Conformal Structures and Holomorphic Quadratic Differentials. Teichmiillers Theorem
4.3 Fenchel-Nielsen Coordinates. An Alternative Approach to the Topology of Teichmiiller Space
4.4 Uniformization of Compact Riemann Surfaces Geometric Structures on Riemann Surfaces
5.1 Preliminaries: Cohomology and Homology Groups
5.2 Harmonic and Holomorphic Differential Forms on Riemann Surfaces
5.3 The Periods of Holomorphic and Meromorphic Differential Forms
5.4 Divisors. The Riemann-Roch Theorem
5.5 Holomorphic 1-Forms and Metrics on Compact Riemann Surfaces
5.6 Divisors and Line Bundles
5.7 Projective Embeddings
5.8 Algebraic Curves
5.9 Abels Theorem and the Jacobi Inversion Theorem
5.10 Elliptic Curves
Bibliography
Index of Notation
Index

前言/序言

　　The present book started from a set of lecture notes for a course taught to stu-dents at an intermediate level in the German system（roughly C0rrespondingto the beginning graduate student level in the US）in the winter term 86／87in Bochum.The original manuscript has been thoroughly reworked severaltimes although its essential aim has not been changed.Traditionally，many graduate courses in mathematics，and in particular thoseon Riemann surface theory，develop their subject in a most systematic，co-herent，and elegant manner from a single point of view and perspective withgreat methodological purity.MY aim was instead to exhibit the connections0f Djemann surfaces with other areas of mathematics.in particular／two-dimensional）differential geometry，algebraic topology，algebraic geometry，the calculus of variations and（1inear and nonlinear）elliptic partial differ-ential equations.I consider Riemann surfaces as an ideal meeting groundfor analysis，geometry，and algebra and as ideally suited for displaying theunity of mathematics.Therefore，they are perfect for introducing intermedi-ate students to advanced mathematics.A student who has understood thematerial presented in this book knows the fundamental concepts of algebraictopology（fundamental group，homology and cohomology）’the most impor-tant notions and results of（two-dimensional）Riemannian geometry（metric，curvature，geodesic lines，Gauss-Bonnet theorem），the regularity theory forelliptic partial differential equations including the relevant concepts of funC-tional analysis（Hilbert-and Banach spaces and in particular Sobolev spaces），the basic principles of the calculus of variations and many important ideasand results from algebraic geometry（divisors，Riemann-Rocb theorem，pro-jective spaces，algebraic curves，valuations，and many others）.Also，she orhe has seen the meaning and the power of all these concepts，methods，andideas at the interesting and nontrivial example of Riemann surfaces.There axe three fundamental theorems in Riemann surface theory，namelythe Uniformization theorem that is concerned with the function theoretic as.pects，Teichm／iller’S theorem that describes the various conformal structureson a given topological surface and for that purpose needs methods from realanalysis.and the Riemann.ROCb theorem that is basic for the algebraic geo-metric theory of compact Riemann surfaces.Among those.

紧黎曼曲面（第3版）（英文版） [Compact Riemann surface:An Introduction to Contemporary Mathemaitce 3rd ed.] 电子书 下载 mobi epub pdf txt

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