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.

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

数学高级讲义ALM

本书可供数学专业的研究生和高年级本科生阅读，也可供相关领域研究人员参考。