Geometry and Analysis(Vol.I)(几何与分析)(第I卷)

This book contains many substantial papers from distinguishedspeakers of a conference 'Geometric Analysis: Present and Future'and an overview of the works of Professor Shing-Tung Yau.Contributors include E. Wit-ten, Y.T. Siu, R. Hamilton, H. Hitchin,B. Lawson, A. Strominger, C. Vafa, W. Schmid, V. Guillemin, N. Mok,D. Christodoulou. This is a valuable reference that gives anup-to-dated summary of geometric analysis and its applications inmany different areas of mathematics.

目录
- FrontMatter
- Part1SummaryofandCommentariesontheWorkofShing-TungYau
- 0CurriculumVitaeofShing-TungYau
- 1ABriefOverviewoftheWorkofShing-TungYauLizhenJi
  - 1Introduction
  - 2AsummaryofsomemajorworksofYau
  - 3TopicsYauhasworkedon
  - 4BasicsonKahler-EinsteinmetricsandCalabiconjectures
  - 5SomeapplicationsofKahler-EinsteinmetricsandCalabi-Yaumanifolds
  - 6Harmonicmaps
  - 7RigidityofKahlermanifolds
  - 8Super-rigidityofspacesofnonpositivecurvature
  - 9SurveypapersbyYau
  - 10OpenproblemsbyYau
  - 11Bookswrittenandco-writtenbyYau
  - 12Bookseditedandco-editedbyYau
  - 13Ph.D.studentsofYau
  - 14PartiallistofpapersandbooksofYau
  - References
- 2Yau'sWorkonFilteringProblemWen-LinChiou,JieHuangandLizhenJi
  - 1Filteringproblem
  - 2Yau'stwomethodsinsolvingnonlinearlteringproblem
    - 2.1Directmethod
    - 2.2Algorithmforrealtimesolutionwithoutmemory
  - References
- 3FromContinuestoDiscrete—Yau'sWorkonGraphTheoryFanChung
- 4Yau'sWorkonModuli,Periods,andMirrorMapsforCalabi-YauManifoldsCharlesF.Doran
  - 1ConstructionofCalabi-Yauthreefolds
  - 2Picard-Fuchsequationsandthemirrormap
  - 3Arithmeticpropertiesofmirrormaps
  - 4PeriodsandmoduliofcomplexToriandK3surfaces
  - References
- 5ReviewonYau'sWorkontheCoupledEinsteinEquationsandtheWaveDynamicsintheKerrGeometryFelixFinster
  - 1CouplingtheEinsteinequationstonon-abeliangaugeeldsandDiracspinors
  - 2ThedynamicsoflinearwavesintheKerrgeometry
  - References
- 6TheWorkofWittenandYauontheAdS/CFTCorrespondenceGregoryJ.Galloway
  - 1Introduction
  - 2TheWitten-YauresultsonAdS/CFT
  - 3Furtherdevelopments
  - References
- 7Yau'sWorkonHeatKernelsAlexanderGrigor'yan
  - 1Thenotionoftheheatkernel
  - 2Estimatingheatkernels
  - 3Someapplicationsoftheheatkernelestimates
  - References
- 8Yau'sContributionstoEngineeringFieldsXianfengDavidGu
  - 1Introduction
  - 2Computationalconformalgeometry
    - 2.1Conformalstructure
    - 2.2Harmonicmap
    - 2.3Surfacericciow
    - 2.4Conformalmappings
    - 2.5Quasi-conformalmappings
    - 2.6Teichmullerspace
  - 3Geometricacquisition
  - 4Computergraphics
  - 5Geometricmodeling
  - 6Medicalimaging
  - 7Computervision
  - 8Wirelesssensornetwork
  - 9Summary
  - References
- 9TheSYZProposalNaichungConanLeung
  - 1Pre-SYZ
  - 2ThebirthofSYZ
  - 3ThegrowingupofSYZ
    - 3.1SpecialLagrangiangeometry
    - 3.2SpecialLagrangianbrations
    - 3.3Affinegeometry
    - 3.4SYZtransformation
  - 4FutureofSYZ
  - References
- 10Yau-ZaslowFormulaNaichungConanLeung
- 11Yau'sWorkonFunctionTheory:HarmonicFunctions,EigenvaluesandtheHeatEquationPeterLi
- 12AVisionofYauonMirrorSymmetryBongLian
  - 1Enumerativegeometry
  - 2GeometryofCalabi-Yaumanifoldsandtheirmodulispaces
  - References
- 13Yau'sWorkonGroupActionsKefengLiu
- 14ChengandYau'sWorkontheMonge-AmpereEquationandAffineGeometryJohnLoftin,Xu-JiaWangandDeaneYang
  - 1Introduction
  - 2TheMonge-Ampereequation
  - 3ChengandYau'sworkontheDirichletproblem
  - 4SubsequentworkontheMonge-Ampereequation
  - 5AffineSpheres
  - 6HyperbolicaffinespheresandrealMonge-Ampereequations
  - 7Afffinemanifolds
  - 8MaximalhypersurfacesinMinkowskispace
  - 9TheMinkowskiproblem
  - 10Convexgeometrywithoutsmoothnessassumptions
    - 10.1Supportfunction
    - 10.2Invariancepropertiesofthesupportfunction
    - 10.3Minkowskisum
    - 10.4Mixedvolume
    - 10.5Surfaceareameasure
    - 10.6Invariancepropertiesofthesurfaceareameasure
    - 10.7TheMinkowskiproblem
    - 10.8TheBrunn-Minkowskiinequality
    - 10.9UniquenessintheMinkowskiproblem
    - 10.10VariationalapproachtotheMinkowskiproblem
  - 11Convexgeometrywithsmoothnessassumptions
    - 11.1TheinverseGaussmap
    - 11.2Theinversesecondfundamentalform
    - 11.3Thecurvaturefunction
    - 11.4Thesurfaceareameasure
    - 11.5TheMinkowskiproblem
    - 11.6TheMinkowskiproblemasaPDE
  - 12ChengandYau'sregularitytheoremfortheMinkowskiproblem
    - 12.1Statement
    - 12.2SketchofProof
  - 13GeneralizationsoftheMinkowskiproblem
  - References
- 15Yau'sWorkonMinimalSurfacesand3-manifoldsFengLuo
- 16TheWorkofSchoenandYauonManifoldswithPositiveScalarCurvatureWilliamMinicozziII
  - 0Introduction
  - 1Topologicalrestrictionsonmanifoldswithpositivescalarcurvature
    - 1.1Stableminimalsurfacesandscalarcurvature
    - 1.2Inductivelyextendingthistohigherdimensions
    - 1.3Preservingpositivescalarcurvatureundersurgery
  - 2Locallyconformallyatmanifolds
    - 2.1Thenewinvariants
    - 2.2Apositivemasstheorem
  - References
- 17Yau'sContributionstoAlgebraicGeometryAndreyTodorov
  - 1Introduction
    - 1.1Yau'sprogram-plenarytalkatICM1982
  - 2Monge-Ampereequationandapplicationstoalgebraicgeometry
    - 2.1SolutionoftheCalabiconjecture
    - 2.2ExistenceofcanonicalmetricsonZariskiopensets
  - 3StablevectorbundlesoverKahlermanifolds
    - 3.1Donaldson-Uhlenbeck-Yautheorem
    - 3.2ApplicationstoKodaira'sclassifficationofsurfaces
  - 4Modulispaces
    - 4.1ExistenceofKahler-EinsteinmetricsondomainofholomorphyandTeichmullerspaces
    - 4.2ModulispacesofK3surfaces
    - 4.3ModulispacesofCYmanifolds
    - 4.4GeneralizationofShwarzlemmabyYauandBaily-Borelcompactification
  - 5ContributionsofYautostringtheory
    - 5.1MirrorsymmetryandSYZconjecture
    - 5.2Largeradiuslimit
    - 5.3Stringtheoryandnumbertheory
    - 5.4RationalcurvesonalgebraicK3surfaces
  - 6Rigidity
    - 6.1Yau'sconjectureaboutrigidityofsomecomplexmanifolds
    - 6.2Geometricproofofmargulis'superrigidity
  - 6.3GeometricproofofKazhdantheoremaboutGaloisconjugationofShimuravarieties
  - References
- 18Yau'sWorkonPositiveMassTheoremsMu-TaoWang
- 19Yau'sConjectureonKahler-EinsteinMetricandStabilityXiaoweiWang
- 20OnYau'sPioneerContributionontheFrankelConjectureandRelatedQuestionsFangyangZheng
- 21Yau'sWorkonInequalitiesBetweenChernNumbersandUniformizationofComplexManifoldsKangZuo
- Part2DifferentialGeometryandDifferentialEquations
- 22GeometryofCompleteGradientShrinkingRicciSolitonsHuai-DongCao
  - 1GradientshrinkingRiccisolitons
  - 2Classificationof3-dimensionalgradientshrinkingsolitons
  - 3Geometryofcompletegradientsolitons
  - References
- 23TheFormationofBlackHolesinGeneralRelativityDemetriosChristodoulou
- 24PageRankasaDiscreteGreen'sFunctionFanChung
  - 1Introduction
  - 2Preliminaries
  - 3Dirichleteigenvalues
  - 4ConnectionsbetweenPageRankanddiscreteGreen'sfunction
  - 5RelatingtheCheegerconstanttothePageRank
  - 6RelatingthePageRankofagraphtothatofitssubgraphs
  - 7ThePageRankandthehittingtime
  - References
- 25AGeodesicEquationintheSpaceofSasakianMetricsPengfeiGuanandXiZhang
- 26SomeInverseSpectralResultsfortheTwo-dimensionalSchrodingerOperatorV.GuilleminandA.Uribe
  - 1Introduction
  - 2TheWeylcalculus
  - 3Somebracketidentities
  - 4ThequantumBirkhoffcanonicalform
  - References
- 27Li-YauEstimatesandtheirHarnackInequalitiesRichardS.Hamilton
  - 1Theheatequation
  - 2TheDirichletproblemfortheheatequation
  - 3Theheatequationintheplane
  - 4Thecastaway
  - 5Endangeredspeciesequation
  - 6Themigrationequation
  - 7Motionofacurvebyitscurvature
  - 8Motionofasurfacebyitsmeancurvature
  - 9MotionofasurfacebyitsGausscurvature
  - References
- 28PlurisubharmonicityinaGeneralGeometricContextF.ReeseHarveyandH.BlaineLawson,Jr
  - 1Introduction
  - 2Geometricallydefinedplurisubharmonicfunctions
  - 3MoregeneralplurisubharmonicfunctionsdefinedbyanellipticconeP+
  - 4P+-plurisubharmonicdistributions
  - 5Upper-semi-continuousP+-plurisubharmonicfunctions
  - 6SomeclassicalfactsthatextendtoP+-plurisubharmonicfunctions
  - 7TheDirichletproblem-uniqueness
  - 8TheDirichletproblem-existence
  - 9P+-convexdomains
  - 10TopologicalrestrictionsonP+-convexdomains
  - 11P+-freesubmanifolds
  - 12P+-convexboundaries
  - References
- 29PoissonModulesandGeneralizedGeometryNigelHitchin
  - 1Introduction
  - 2Poissonmodules
    - 2.1Definitions
    - 2.2Aconstruction
  - 3TheSerreconstruction
    - 3.1Thealgebraicapproach
    - 3.2Theanalyticalapproach
    - 3.3Thesecondsection
  - 4Generalizedgeometry
    - 4.1Basicfeatures
    - 4.2GeneralizedDolbeaultoperators
    - 4.3Thecanonicalbundle
  - 5Ageneralizedconstruction
    - 5.1Theproblem
    - 5.2Generalizedcomplexsubmanifolds
    - 5.3Theconstruction
  - 6Anapplication
  - References
- 30UniquenessofSolutionstoMeanFieldEquationsofLiouvilleTypeinTwo-dimensionChang-shouLin
  - 1Introduction
  - 2UniquenessinR2
  - 3UniquenessinboundeddomainsofR2
  - 4Onofriinequalityanditsgeneralization
  - 5MeanfieldequationandGreenfunctionsontorus
  - 6GeneralizedLiouvillesystem
  - References
- 31MonotonicityandHolomorphicFunctionsLeiNi
- 32DecayofSolutionstotheCauchyProblemintheKerrGeometryforVariousPhysicalSystems:StabilityofBlackHolesJ.A.Smoller
  - 1Introduction
  - 2Main
  - References
- 33TheCalabi-YauEquation,SymplecticFormsandAlmostComplexStructuresValentinoTosattiandBenWeinkove
  - 1Background-Yau'sTheorem
  - 2Donaldson'sconjectureandapplications
  - 3EstimatesfortheCalabi-Yauequation
  - 4Methods
  - 5Amonotonicityformula
  - References
- 34UnderstandingWeil-PeterssonCurvatureScottA.Wolpert
  - 1Introduction
  - 2BasicsofTeichmullertheory
  - 3WPintrinsicgeometry
  - 4Methods
  - 5Applicationsofcurvature
  - 5.1TheworkofLiu,SunandYau
  - 5.2Themodelmetric4dr2+r6d2
  - 5.3Projectionanddistancetoastratum
  - References
- 35ExamplesofPositivelyCurvedCompleteKahlerManifoldsHung-HsiWuandFangyangZheng
  - 1Introduction
  - 2TheABCDfunctions
  - 3Characterizationbythefunction
  - 4Someexamples
  - 5Characterizationbysurfaceofrevolution
  - 6CorrelationbetweenvolumegrowthandcurvatureDecay
  - References

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