几何分析手册（第2卷）(精装，未翻阅，近似全新，1版1次，英文版)

　　GeometricAnalysiscombinesdifferentialequationsanddifferentialgeometry.Animportantaspectistosolvegeometricproblemsbystudyingdifferentialequations.BesidessomeknownlineardifferentialoperatorssuchastheLaplaceoperator,manydifferentialequationsarisingfromdifferentialgeometryarenonlinear.AparticularlyimportantexampleistheIVlonge-Ampereequation;Applicationstogeometricproblemshavealsomotivatednewmethodsandtechniquesindifferen-rialequations.Thefieldofgeometricanalysisisbroadandhashadmanystrikingapplications.Thishandbookofgeometricanalysisprovidesintroductionstoandsurveysofimportanttopicsingeometricanalysisandtheirapplicationstorelatedfieldswhichisintendtobereferredbygraduatestudentsandresearchersinrelatedareas.

HeatKernelsonMetricMeasureSpaceswithRegularVolumeGrowth
AlexanderGriqoryan
1Introduction
1.1HeatkernelinRn
1.2HeatkernelsonRiemannianmanifolds
1.3HeatkernelsoffractionalpowersofLaplacian
1.4Heatkernelsonfractalspaces
1.5Summaryofexamples
2Abstractheatkernels
2.1Basicdefinitions
2.2TheDirichletform
2.3Identifyinginthenon-localcase
2.4Volumeofballs
3Besovspaces
3.1BesovspacesinRn
3.2Besovspacesinametricmeasurespace
3.3EmbeddingofBesovspacesintoHSlderspaces.
4Theenergydomain
4.1Alocalcase
4.2Non-localcase
4.3Subordinatedheatkernel
4.4Besselpotentialspaces
5Thewalkdimension
5.1Intrinsiccharacterizationofthewalkdimension
5.2Inequalitiesforthewalkdimension
6Two-sidedestimatesinthelocalcase
6.1TheDirichletforminsubsets
6.2Maximumprinciples
6.3Atailestimate
6.4Identifyinginthelocalcase
References
AConvexityTheoremandReducedDelzantSpacesBongH.Lian，BailinSong
1Introduction
2Convexityofimageofmomentmap
3Rationalityofmomentpolytope
4RealizingreducedDelzantspaces
5ClassificationofreducedDelzantspaces
References
LocalizationandsomeRecentApplications
BongH.Lian，KefengLiu
1Introduction
2Localization
3Mirrorprinciple
4Hori-Vafaformula
5TheMarino-VafaConjecture
6Twopartitionformula
7Theoryoftopologicalvertex
8Gopakumar-Vafaconjectureandindicesofellipticoperators..
9TwoproofsoftheELSVformula
10AlocalizationproofoftheWittenconjecture
11Finalremarks
References
Gromov-WittenInvariantsofToricCalabi-YauThreefoldsChiu-ChuMelissaLiu
1Gromov-WitteninvariantsofCalabi-Yau3-folds
1.1SymplecticandalgebraicGromov-Witteninvariants
1.2Modulispaceofstablemaps
1.3Gromov-WitteninvariantsofcompactCalabi-Yau3-folds
1.4Gromov-WitteninvariantsofnoncompactCalabi-Yau3-folds
2Traditionalalgorithminthetoriccase
2.1Localization
2.2Hodgeintegrals
3Physicaltheoryofthetopologicalvertex
4Mathematicaltheoryofthetopologicalvertex
4.1Locallyplanartrivalentgraph
4.2FormaltoricCalabi-Yau(FTCY)graphs
4.3Degenerationformula
4.4Topologicalvertex"
4.5Localization
4.6Framingdependence
4.7Combinatorialexpression
4.8Applications
4.9Comparison
5GW/DTcorrespondencesandthetopologicalvertex
Acknowledgments
References
SurveyonAffineSpheres
JohnLoftin
1Introduction
2Affinestructureequations
3Examples
4Two-dimensionalaffinespheresandTiteicasequation
5Monge-Ampreequationsandduality
6Globalclassificationofaffinespheres
7Hyperbolicaffinespheresandinvariantsofconvexcones
8Projectivemanifolds
9Affinemanifolds
10Affinemaximalhypersurfaces
11Affinenormalflow
References
ConvergenceandCollapsingTheoremsinRiemannianGeometry
XiaochunRong
Introduction
1Gromov-Hausdorffdistanceinspaceofmetricspaces
1.1TheGromov-Hausdorffdistance
1.2Examples
1.3AnalternativeformulationofGH-distance
1.4Compactsubsetsof(Met，dGH)
1.5EquivariantGH-convergence
1.6PointedGH-convergence
2Smoothlimits-fibrations
2.1Thefibrationtheorem
2.2Sectionalcurvaturecomparison
2.3Embeddingviadistancefunctions
2.4Fibrations
2.5Proofoftheorem2.1.1
2.6Centerofmass
2.7Equivariantfibrations
2.8Applicationsofthefibrationtheorem
3Convergencetheorems
3.1Cheeger-Gromovsconvergencetheorem
3.2Injectivityradiusestimate
3.3Someellipticestimates
3.4Harmonicradiusestimate
3.5Smoothingmetrics
4Singularlimits-singularfibrations
4.1Singularfibrations
4.2Controlledhomotopystructurebygeometry
4.3The∏2-finitenesstheorem
4.4Collapsedmanifoldswithpinchedpositivesectionalcurvature
5Almostflatmanifolds
5.1Gromovstheoremonalmostflatmanifolds
5.2TheMargulislemma
5.3Flatconnectionswithsmalltorsion
5.4Flatconnectionwithaparalleltorsion
5.5Proofs——partI
5.6Proofs——partII
5.7Refinedfibrationtheorem
References
GeometricTransformationsandSolitonEquations
Chuu-LianTerng"
1Introduction
2Themovingframemethodforsubmanifolds
3LinecongruencesandBacklundtransforms
4SpherecongruencesandRibaucourtransforms
5Combescuretransforms，O-surfaces，andk-tuples
6FrommovingframetoLaxpair
7Solitonhierarchiesconstructedfromsymmetricspaces
8TheU-systemandtheGauss-Codazziequations
9Loopgroupactions
10Actionofsimpleelementsandgeometrictransforms
References
AffineIntegralGeometryfromaDifferentiableViewpoint
DeaneYang
1Introduction
2Basicdefinitionsandnotation
2.1Lineargroupactions
3Objectsofstudy
3.1Geometricsetting
3.2Convexbody
3.3Thespaceofallconvexbodies
3.4Valuations
4Overallstrategy
5Fundamentalconstructions
5.1Thesupportfunction
5.3Thepolarbody
5.4TheinverseGaussmap
5.5Thesecondfundamentalform
5.6TheLegendretransform
5.7ThecurvaturefunctionThehomogeneouscontourintegral
6.1Homogeneousfunctionsanddifferentialforms
6.2Thehomogeneouscontourintegralforadifferentialform
6.3Thehomogeneouscontourintegralforameasure
6.4Homogeneousintegralcalculus
7Anexplicitconstructionofvaluations
7.1Duality
7.2Volume
8Classificationofvaluations
9Scalarvaluations
9.1SL(n)-invariantvaluations
9.2Hugstheorem
10ContinuousGL(n)-homogeneousvaluations
10.1Scalarvaluations
10.2Vector-valuedvaluations
11Matrix-valuedvaluations.
11.1TheCramer-Raoinequality
12Homogeneousfunction-andconvexbody-valuedvaluations.
13Questions
References
ClassificationofFakeProjectivePlanes
Sai-KeeYeung
1Introduction
2Uniformizationoffakeprojectiveplanes
3Geometricestimatesonthenumberoffakeprojectiveplanes.
4Arithmeticityoflatticesassociatedtofakeprojectiveplanes.
5CovolumeformulaofPrasad
6Formulationofproof
7Statementsoftheresults
8Furtherstudies
References