幾何與分析中的最新進展（簡體書）

《幾何與分析中的最新進展》彙集了微分幾何、幾何分析、微分方程等方面一些綜述性文章或最新進展研究成果，特別有益於幾何、分析專業的碩士生、博士生以及剛剛開始從事這些領域研究的數學工作者，對在相關領域工作的其他數學工作者也很有參考價值。．

《幾何與分析中的最新進展》由高等教育出版社出版。

MinimalVolumeandSimplicialVolumeofVisibilityn-manifoldsand
All3-manifolds
1Introduction
2SimplicialVolumeandBoundedCohomology
3VisibilityManifoldsandGromov-hyperbolicSpaces
4Gromov'sSimplicialVolumeofVisibilityn，-manifoldsand
Compact3-manifolds.
References

RigidityTheoremsforLagrangianSubmanifoldsofComplexSpaceFormswithConformalMaslovForm
1Introduction
2Preliminaries.
3RigidityTheoremsforLagrangianSubmanifolds
References

MethodofMovingPlanesinIntegralFormsandRegularityLifting
1Introduction
2IllustrationofMMPinIntegralForms
3VariousApplicationsofMMPinIntegralForms.
4RegularityLiftingbyContractingOperators.
5RegularityLiftingbyCombinationsofContractingand
ShrinkingOperators
References

TheRicciCurvatureinFinslerGeometryDefinitionsandNotations
2RicciCurvatureofRandersMetrics.
3VolumeComparisoninFinslerGeometry
4TheRoleoftheRicciCurvatureinProjectiveGeometry

SpecificNon-KahlerHermitianMetricsonCompactComplex
Manifolds
1Introduction
2BalancedMetricsUndertheConifoldTrlansition
3TheSupersymmetricSolutions
4TheForm-typeCalabi-YauEquation
5TheGeneralizedGauduchonMetrics
References.
OntheO2-scalarCurvatureandItsApplications
1Introduction
2O2-YamabeProblem
3AQuotientEquation.
4EllipticityofaQuotientEquationanda3-dimensional
SphereTheorem
5ARoughClassificationofMetricsofPositiveScalarCurvature
6AnAlmostSchurTheorem
References.

IsometricEmbeddingofSurfacesinR3
Introduction
2LocallsometricEmbeddingofSurfaces.
3GlobalIsometricEmbeddingofSurfaces.
References

TheLagrangianMeanCurvatureFlowAlongtheKahler-RicciFlow
1Introduction，
2LagrangianPropertyIsPreserved
References
OnMagnetohydrodynamicswithPartialMagneticDissipation
nearEquilibrium
Introduction
2ReformulationoftheIdealMHD
3GlobalExistence
4Acknowledgement


HyperbolicGradientFlow：EvolutionofGraphsinRn+1
TheMoser-TrudingerandAdamsInequalitiesandEllipticandSubellipticEquationswithNonlinearityofExponentialGrowth
NavigationProblemandRandersMetrics
LorentzianIsoparametricHypersurfacesintheLorentzianSpheres（n+1）1
AsymptoticallyHyperbolicManifoldsandConformalGeometry
ACharacterizationofRandersMetricsofScalarFlagCurvature
PrescribedWeingartenCurvatureEquations
GeometryProblemsRelatedwithQuasi-localMassinGeneralRelativity
Concerningthe/4NormsofTypicalEigenfunctionsonCompactSurfaces
AnalysisonRiemannianManifoldswithNon-convexBoundary
TheUnityofp-harmonicGeometry
HermitianHarmonicMapsBetweenAlmostHermuitianManifolds
AGlobalMeanValueInequalityforPlurisubharmonicFunctions
onaCompactKahlerManifold
AListofPublicationsbyProfessorZhengguoBai
AListofPublicationsbyProfessorYibingShen
AListofGraduateStudentsofProfessorsBaiandShen．

5 A Rough Classification of Metrics of PositiveScalar Curvature
We hope to use σ2-scalar curvature to give a further classification of the manifoldsadmitting metric with positive scalar curvature.For the further discussion let USfirst recall the following definition.
(1+)Closed connected manifolds with a Riemannian metric whose scalar curva-ture is nonnegative and not identically 0.
(10)Closed connected manifolds with a Riemannian metric with nonnegativescalar curvature,but not in class(1+).
(1_)Closed connected manifolds not in classes(1+)or(10).
Theorem A.(Trichotomy Theorem[49,50])Let Mn be a closed connected man—ifold of dimension n≥3.
(1)If M belongs to class(1+),then every smooth function is the scalar curvaturefunction for some Riemannian metric on M.
(2)If M belongs to class(10),then a smooth function f is the scalar curvaturefunction of some Riemannian metric on M if and only if f(x)<0,dr somepoint="" x∈m.or="" else="" f="0.If" the="" scalar="" curvature="" of="" some="" g="" vanishesidentically,then="" g="" is="" ricci="" flat.="">
(3)If M belongs to class(1-),then a smooth function f is the scalar curvaturefunction of some Riemannian metric on M if and only if f(x)<0 for="" somepoint="" x∈m.="">
From Theorem A or an earlier result of Aubin [2] we know that a negativefunction can always be realized as a scalar function of a metric.See also the resultsof [17] and [57] for the existence of negative Ricci curvature.The class of (10)isvery small and consisting of very special manifolds.thanks to a result of Futaki[15].Theorem A also implies that class(1+)is just the class of manifolds whichadmits a metric of positive scalar curvature.There are topological obstructions for the manifolds with positive scalar curvature,see[55] and[43].This class attractsmuch attention of geometers for many)rears,especially after the work of Gromov—Lawsonl241 and Schoen—Yau [61].The most important problem in this field is the Gromov—Lawson—Rosenberg conjecture which was proved by Stolz [64] in thesimply connected case.For this conjecture see for nstance [59] and [66].