几何分析手册（第Ⅰ卷）（英文）
出版时间:
2008-08
版次:
1
ISBN:
9787040252880
定价:
128.00
装帧:
精装
开本:
16开
纸张:
胶版纸
页数:
676页
字数:
820千字
正文语种:
英语
26人买过
-
GeometricAnalysiscombinesdifferentialequationsanddifferentialgeometry.Animportantaspectistosolvegeometricproblemsbystudyingdifferentialequations.BesidessomeknownlineardifferentialoperatorssuchastheLaplaceoperator，manydifferentialequationsarisingfromdifferentialgeometryarenonlinear.AparticularlyimportantexampleistheMonge-Ampereequation.Applicationstogeometricproblemshavealsomotivatednewmethodsandtechniquesindifferen-tialequations.Thefieldofgeometricanalysisisbroadandhashadmanystrikingapplications.Thishandbookofgeometricanalysisprovidesintroductionstoandsurveysofimportanttopicsingeometricanalysisandtheirapplicationstorelatedfieldswhichisintendtobereferredbygraduatestudentsandresearchersinrelatedareas. NumericalApproximationstoExtremalMetricsonToricSurfaces
1Introduction
2Theset-up
2.1Algebraicmetrics
2.2Decompositionofthecurvaturetensor
2.3Integration
3Numericalalgorithms：balancedmetricsandrefinedapproximations
4Numericalresults
4.1Thehexagon
4.2Thepentagon
4.3Theoctagon
4.4Theheptagon
5Conclusions
References
KahlerGeometryonToricManifolds,andsomeotherManifoldswithLargeSymmetry
Introduction
1Background
1.1Gaugetheoryandholomorphicbundles
1.2Symplecticandcomplexstructures
1.3Theequations
2Toricmanifolds
2.1Localdifferentialgeometry
2.2Theglobalstructure
2.3Algebraicmetricsandasymptotics
2.4Extremalmetricsontoricvarieties
3ToricFanomanifolds
3.1TheKahler-Riccisolitonequation
3.2Continuitymethod,convexityandafundamentalinequality
3.3Aprioriestimate
3.4ThemethodofWangandZhu
4Variantsoftoricdifferentialgeometry
4.1Multiplicity-freemanifolds
4.2Manifoldswithadenseorbit
5TheMukai-Umemuramanifoldanditsdeformations
5.1Mukai'sconstruction
5.2Topologicalandsymplecticpicture
5.3Deformations
5.4Thea-invariant
References
GluingConstructionsofSpecialLagrangianCones
1Introduction
2SpecialLagrangianconesandspecialLegendriansubmanifoldsofS2n-1
3CohomogeneityonespecialLegendriansubmanifoldsofS2n-1
4ConstructionoftheinitialalmostspecialLegendriansubmanifolds
5Thesymmetrygroupandthegeneralframeworkforcorrectingtheinitialsurfaces
6Thelinearizedequation
7UsingtheGeometricPrincipletoprescribetheextendedsubstitutekernel
8Themainresults
ASymmetriesandquadratics
References
HarmonicMappings
1Introduction
2HarmonicmappingsfromtheperspectiveofRiemanniangeometry
2.1HarmonicmappingsbetweenRiemannianmanifolds：definitionsandproperties
2.2Theheatflowandharmonicmappingsintononpositivelycurvedmanifolds
2.3HarmonicmappingsintoconvexregionsandapplicationstotheBernsteinproblem
3Harmonicmappingsfromtheperspectiveofabstractanalysisandconvexitytheory
3.1Existence
3.2Regularity
3.3Uniquenessandsomeapplications
4HarmonicmappingsinKahlerandalgebraicgeometry
4.1Rigidityandsuperrigidity
4.2Harmonicmapsandgrouprepresentations
4.3Kahlergroups
4.4Quasiprojectivevarietiesandharmonicmappingsofinfiniteenergy
5HarmonicmappingsandRiemannsurfaces
5.1FamiliesofRiemannsurfaces
……
HarmonicFunctionsonCompleteRiemannianManifolds
ComplexityofSolutionsofPartialDifferentialEquations
VariationalPrinciplesonTriangulatedSurfaces
AsymptoticStructuresintheGeometryofStabilityandExtremalMetrics
StableConstantMeanCurvatureSurfaces
AGeneralAsymptoticDecayLemmaforEllipticProblems
UniformizationofOpenNonnegativelyCurvedK／ihlerManifoldsinHigherDimensions
GeometryofMeasures：HarmonicAnalysisMeetsGeometricMeasureTheory
TheMongeAmpereEequationanditsGeometricAapplications
LecturesonMeanCurvatureFlowsinHigherCodimensions
LocalandGlobalAnalysisofEigenfunctionsonRiemannianManifolds
Yau’SFormofSchwarzLemmaandArakelovInequalityOnModuliSpacesofProjectiveManifolds
-
内容简介:
GeometricAnalysiscombinesdifferentialequationsanddifferentialgeometry.Animportantaspectistosolvegeometricproblemsbystudyingdifferentialequations.BesidessomeknownlineardifferentialoperatorssuchastheLaplaceoperator，manydifferentialequationsarisingfromdifferentialgeometryarenonlinear.AparticularlyimportantexampleistheMonge-Ampereequation.Applicationstogeometricproblemshavealsomotivatednewmethodsandtechniquesindifferen-tialequations.Thefieldofgeometricanalysisisbroadandhashadmanystrikingapplications.Thishandbookofgeometricanalysisprovidesintroductionstoandsurveysofimportanttopicsingeometricanalysisandtheirapplicationstorelatedfieldswhichisintendtobereferredbygraduatestudentsandresearchersinrelatedareas.
-
目录:
NumericalApproximationstoExtremalMetricsonToricSurfaces
1Introduction
2Theset-up
2.1Algebraicmetrics
2.2Decompositionofthecurvaturetensor
2.3Integration
3Numericalalgorithms：balancedmetricsandrefinedapproximations
4Numericalresults
4.1Thehexagon
4.2Thepentagon
4.3Theoctagon
4.4Theheptagon
5Conclusions
References
KahlerGeometryonToricManifolds,andsomeotherManifoldswithLargeSymmetry
Introduction
1Background
1.1Gaugetheoryandholomorphicbundles
1.2Symplecticandcomplexstructures
1.3Theequations
2Toricmanifolds
2.1Localdifferentialgeometry
2.2Theglobalstructure
2.3Algebraicmetricsandasymptotics
2.4Extremalmetricsontoricvarieties
3ToricFanomanifolds
3.1TheKahler-Riccisolitonequation
3.2Continuitymethod,convexityandafundamentalinequality
3.3Aprioriestimate
3.4ThemethodofWangandZhu
4Variantsoftoricdifferentialgeometry
4.1Multiplicity-freemanifolds
4.2Manifoldswithadenseorbit
5TheMukai-Umemuramanifoldanditsdeformations
5.1Mukai'sconstruction
5.2Topologicalandsymplecticpicture
5.3Deformations
5.4Thea-invariant
References
GluingConstructionsofSpecialLagrangianCones
1Introduction
2SpecialLagrangianconesandspecialLegendriansubmanifoldsofS2n-1
3CohomogeneityonespecialLegendriansubmanifoldsofS2n-1
4ConstructionoftheinitialalmostspecialLegendriansubmanifolds
5Thesymmetrygroupandthegeneralframeworkforcorrectingtheinitialsurfaces
6Thelinearizedequation
7UsingtheGeometricPrincipletoprescribetheextendedsubstitutekernel
8Themainresults
ASymmetriesandquadratics
References
HarmonicMappings
1Introduction
2HarmonicmappingsfromtheperspectiveofRiemanniangeometry
2.1HarmonicmappingsbetweenRiemannianmanifolds：definitionsandproperties
2.2Theheatflowandharmonicmappingsintononpositivelycurvedmanifolds
2.3HarmonicmappingsintoconvexregionsandapplicationstotheBernsteinproblem
3Harmonicmappingsfromtheperspectiveofabstractanalysisandconvexitytheory
3.1Existence
3.2Regularity
3.3Uniquenessandsomeapplications
4HarmonicmappingsinKahlerandalgebraicgeometry
4.1Rigidityandsuperrigidity
4.2Harmonicmapsandgrouprepresentations
4.3Kahlergroups
4.4Quasiprojectivevarietiesandharmonicmappingsofinfiniteenergy
5HarmonicmappingsandRiemannsurfaces
5.1FamiliesofRiemannsurfaces
……
HarmonicFunctionsonCompleteRiemannianManifolds
ComplexityofSolutionsofPartialDifferentialEquations
VariationalPrinciplesonTriangulatedSurfaces
AsymptoticStructuresintheGeometryofStabilityandExtremalMetrics
StableConstantMeanCurvatureSurfaces
AGeneralAsymptoticDecayLemmaforEllipticProblems
UniformizationofOpenNonnegativelyCurvedK／ihlerManifoldsinHigherDimensions
GeometryofMeasures：HarmonicAnalysisMeetsGeometricMeasureTheory
TheMongeAmpereEequationanditsGeometricAapplications
LecturesonMeanCurvatureFlowsinHigherCodimensions
LocalandGlobalAnalysisofEigenfunctionsonRiemannianManifolds
Yau’SFormofSchwarzLemmaandArakelovInequalityOnModuliSpacesofProjectiveManifolds
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