几何分析手册（第3卷）（英文版）

图书标准信息

• 作者 季理真 著；[美]季理真 编
• 出版社 高等教育出版社
• 出版时间 2010-02
• 版次 1
• ISBN 9787040288841
• 定价 88.00元
• 装帧 精装
• 开本 16开
• 纸张 铜版纸
• 页数 472页
• 字数 580千字
• 正文语种 英语

【内容简介】

GeometricAnalysiscombinesdifferentialequationsanddifferentialgeometry.Animportantaspectistosolvegeometricproblemsbystudyingdifferentialequations.BesidessomeknownlineardifferentialoperatorssuchastheLaplaceoperator，manydifferentialequationsarisingfromdifferentialgeometryarenonlinear.AparticularlyimportantexampleistheMonge-Ampereequation.Applicationstogeometricproblemshavealsomotivatednewmethodsandtechniquesindifferentialequations.Thefieldofgeometricanalysisisbroadandhashadmanystrikingapplications.Thishandbookofgeometricanalysisprovidesintroductionstoandsurveysofimportanttopicsingeometricanalysisandtheirapplicationstorelatedfieldswhichisintendtobereferredbygraduatestudentsandresearchersinrelatedareas.

【目录】

ASurveyofEinsteinMetricson4-manifolds
MichaelT.Anderson
1Introduction
2Briefreview：4-manifolds，complexsurfacesandEinsteinmetrics
3ConstructionsofEinsteinmetricsI
4ObstructionstoEinsteinmetrics
5ModulispacesI
6ModuⅡspacesⅡ
7ConstructionsofEinsteinmetricsⅡ
8Concludingremarks
References

SphereTheoremsinGeometry
SimonBrendle，RichardSchoen
1TheTopologicalSphereTheorem
2Manifoldswithpositiveisotropiccurvature
3TheDifferentiableSphereTheorem
4NewinvariantcurvatureconditionsfortheRicciflow
5Rigidityresultsandtheclassificationofweakly1/4-pinchedmanifolds
6HamiltonsdifferentialHarnackinequalityfortheRicciflow
7Compactnessofpointwisepinchedmanifolds
References

CurvatureFlowsandCMCHypersurfaces
ClausGerhardt
1Introduction
2Notationsandpreliminaryresults
3Evolutionequationsforsomegeometricquantities.
4Essentialparabolicflowequations
5Existenceresults
6CurvatureflowsinRiemannianmanifolds
7FoliationofaspacetimebyCMChypersurfaces
8TheinversemeancurvatureflowinLorentzianspaces
References

GeometricStructuresonRiemannianManifolds
NaichungConanLeung
1Introduction
2Topologyofmanifolds
2.1Cohomologyandgeometryofdifferentialforms
2.2Hodgetheorem
2.3Witten-Morsetheory
2.4Vectorbundlesandgaugetheory
3Riemanniangeometry
3.1TorsionandLevi-Civitaconnections
3.2ClassificationofRiemannianholonomygroups
3.3Riemanniancurvaturetensors
3.4Flattori
3.5Einsteinmetrics
3.6Minimalsubmanifolds
3.7Harmonicmaps
4Orientedfourmanifolds
4.1Gaugetheoryindimensionfour
4.2Riemanniangeometryindimensionfour
5Kaihlergeometry
5.1Kahlergeometry——complexaspects
5.2Kahlergeometry——Riemannianaspects
5.3Kahlergeometry——symplecticaspects
5.4Gromov-Wittentheory
6Calabi-Yaugeometry
6.1Calabi-Yaumanifolds
6.2SpecialLagrangiangeometry
6.3Mirrorsymmetry
6.4K3surfaces
7Calabi-Yau3-folds
7.1ModuliofCYthreefolds
7.2CurvesandsurfacesinCalabi-Yauthreefolds
7.3Donaldson-ThomasbundlesoverCalabi-Yauthreefolds.
7.4SpecialLagrangiansubmanifoldsinCY3
7.5MirrorsymmetryforCalabi-Yauthreefolds
8G2-geometry
8.1G2-manifolds
8.2ModuliofG2-manifolds
8.3（Co-）associativegeometry
8.4G2-Donaldson-Thomasbundles
8.5G2-dualities，trialitiesandM-theory
9Geometryofvectorcrossproducts
9.1VCPmanifolds
9.2Instantonsandbranes
9.3Symplecticgeometryonhigherdimensionalknotspaces.
9.4C-VCPgeometry
9.5HyperkahlergeometryonisotropicknotspacesofCY
10Geometryovernormeddivisionalgebras
10.1Manifoldsovernormedalgebras
10.2Gaugetheoryover（special）A-manifolds
10.3A-submanifoldsand（special）Lagrangiansubmanifolds.
11Quaterniongeometry
11.1Hyperkahlergeometry
11.2Quaternionic-Kahlergeometry
12Conformalgeometry
13GeometryofRiemanniansymmetricspaces
13.1Riemanniansymmetricspaces
13.2Jordanalgebrasandmagicsquare
13.3Hermitianandquaternionicsymmetricspaces
14Conclusions
References

SymplecticCalabi-YauSurfaces
Tian-JunLi
1Introduction
2Linearsymplecticgeometry
2.1Symplecticvectorspaces
2.2Compatiblecomplexstructures
2.3Hermitianvectorspaces
2.44-dimensionalgeometry
3Symplecticmanifolds
3.1Almostsymplecticandalmostcomplexstructures
3.2Cohomologicalinvariantsandspaceofsymplecticstructures
3.3MoserstabilityandDarbouxcharts
3.4Submanifoldsandtheirneighborhoods
3.5Constructions
4AlmostKahlergeometry
4.1AlmostHermitianmanifolds，integrabilityandoperators.
4.2Levi-Civitaconnection
4.3ConnectionsandcurvatureonHermitianbundles
4.4ChernconnectionandHermitiancurvatures
4.5Theself-dualoperator
4.6Diracoperators
4.7WeitzenbSckformulasandsomealmostKahleridentities.
5Seiberg-Wittentheory-threefacets
5.1SWequations
5.2Pin（2）symmetryforaspinreduction
5.3ThecompactnessandHausdorffpropertyofthemodulispace
5.4Genericsmoothnessofthemodulispace
5.5Furutasfinitedim.Approximations
5.6SWinvariants
5.7SymplecticSWequationsandTaubesnonvanishing
5.8SymplecticSWsolutionsandPseudo-holomorphiccurves.
5.9BordismSWinvariantsviafinitedim.Approximations
5.10Mod2vanishingandhomologytype
6SymplecticCalabi-Yauequation
6.1Uniquenessandopenness
6.2Aprioriestimates
7SymplecticCalabi-Yausurfaces
7.1SymplecticbirationalgeometryandKodairadimension
7.2Examples
7.3Homologiealclassification
7.4Furtherquestions
References

LecturesonStabilityandConstantScalarCurvature
D.H.Phong，Jacob$turm
1Introduction
2TheconjectureofYau
2.1ConstantscalarcurvaturemetricsinagivenKahlerclass.
2.2ThespecialcaseofKahler-Einsteinmetrics
2.3TheconjectureofYau
3Theanalyticproblem
3.1Fourthordernon-linearPDEandMonge-Ampereequations
3.2Geometricheatflows
3.3Variationalformulationandenergyfunctionals
4ThespacesKkofBergmanmetrics
4.1Kodairaimbeddings
4.2TheTian-Yau-Zelditchtheorem
5ThefunctionalF0ω0onKk
5.1F0ω0andbalanceimbeddings
5.2F0ω0andtheEuler-LagrangeequationR-R=0
5.3F0ω0andMonge-Amperemasses
6Notionsofstability
6.1StabilityinGIT
6.2Donaldsonsinfinite-dimensionalGIT
6.3StabilityconditionsonDiff（X）orbits
7Thenecessityofstability
7.1TheMoser-TrudingerinequalityandanalyticK-stability
7.2NecessityofChow-Mumfordstability
7.3NecessityofsemiK-stability
8Sufficientconditions：theKⅡhler-Einsteincase
8.1Theα-invariant
8.2Nadelsmultiplieridealsheavescriterion
8.3TheKahler-Ricciflow
9GeneralL：energyfunctionalsandChowpoints
9.1F0ωandChowpoints
9.2KwandChowpoints
10GeneralL：theCalabienergyandtheCalabiflow
10.1TheCalabiflow
10.2Extremalmetricsandstability
11GeneralL：toricvarieties
11.1Symplecticpotentials
11.2K-stabilityontoricvarieties
11.3TheK-unstablecase
12Geodesicsinthespace/gofKaihlerpotentials
12.1TheDirichletproblemforthecomplexMonge-Ampereequation
12.2Methodofellipticregularizationandaprioriestimates
12.3Geodesicsin/gandgeodesicsin/gk
References

AnalyticAspectofHamiltonsRicciFlow
Xi-PingZhu
Introduction
1Short-timeexistenceanduniqueness
2Curvatureestimates
2.1Shislocalderivativeestimates
2.2Preservingpositivecurvature
2.3Hamilton-Iveypinchingestimate
2.4Li-Yau-Hamiltoninequality
3Singularitiesofsolutions
3.1Canalltypesofsingularitiesbeformed
3.2Singularitymodels
3.3Canonicalneighborhoodstructure
4Longtimebehaviors
4.1TheRicciflowontwo-manifolds
4.2TheRicciflowonthree-manifolds
4.3DifferentialSphereTheorems
References