紧黎曼曲面：第3版

Preface

1TopologicalFoundations

1.1ManifoldsandDifferentiableManifolds

1.2HomotopyofMaps.TheFundamentalGroup

1.3Coverings

1.4GlobalContinuationofFunctionsonSimply-ConnectedManifolds

2DifferentialGeometryofRiemannSurfaces

2.1TheConceptofaRiemannSurface

2.2SomeSimplePropertiesofRiemannSurfaces

2.3MetricsonRiemannSurfaces

2.3ATriangulationsofCompactRiemannSurfaces

2.4DiscreteGroupsofHyperbolicIsometries.FundamentalPolygons.SomeBasicConceptsofSurfaceTopologyandGeometry.

2.4ATheTopologicalClassificationofCompactRiemannSurfaces

2.5TheTheoremsofGauss-BonnetandRiemann-Hurwitz

2.6AGeneralSchwarzLemma

2.7ConformalStructuresonTori

3HarmonicMaps

3.1Review：BanachandHilbertSpaces.TheHilbertSpaceL2

3.2TheSobolevSpaceW12=H12

3.3TheDirichletPrinciple.WeakSolutionsofthePoissonEquation

3.4HarmonicandSubharmonicFunctions

3.5TheCaRegularityTheory

3.6MapsBetweenSurfaces.TheEnergyIntegral.DefinitionandSimplePropertiesofHarmonicMaps

3.7ExistenceofHarmonicMaps

3.8RegularityofHarmonicMaps

3.9UniquenessofHarmonicMaps

3.10HarmonicDiffeomorphisms

3.11MetricsandConformalStructures

4TeichmullerSpaces

4.1TheBasicDefinitions

4.2HarmonicMaps,ConformalStructuresandHolomorphicQuadraticDifferentials.TeichmiillersTheorem

4.3Fenchel-NielsenCoordinates.AnAlternativeApproachtotheTopologyofTeichmiillerSpace

4.4UniformizationofCompactRiemannSurfaces

5GeometricStructuresonRiemannSurfaces

5.1Preliminaries：CohomologyandHomologyGroups

5.2HarmonicandHolomorphicDifferentialFormsonRiemannSurfaces

5.3ThePeriodsofHolomorphicandMeromorphicDifferentialForms

5.4Divisors.TheRiemann-RochTheorem

5.5Holomorphic1-FormsandMetricsonCompactRiemannSurfaces

5.6DivisorsandLineBundles

5.7ProjectiveEmbeddings

5.8AlgebraicCurves

5.9AbelsTheoremandtheJacobiInversionTheorem

5.10EllipticCurves

Bibliography

IndexofNotation

Index

　　本书由世界图书出版公司出版，主要内容包括：UniformizationofCompactRiemannSurfacesGeometricStructuresonRiemannSurfaces、Preliminaries：CohomologyandHomologyGroups、HarmonicandHolomorphicDifferentialFormsonRiemannSurfaces、ThePeriodsofHolomorphicandMeromorphicDifferentialForms、Divisors.TheRiemann-RochTheorem、Holomorphic1-FormsandMetricsonCompactRiemannSurfaces、DivisorsandLineBundles等。