几何与分析中的最新进展

出版时间：2012-6  出版社：东瑜昕、 等 高等教育出版社 (2012-06出版)  作者：东瑜昕  页数：522  
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内容概要

　　《几何与分析中的最新进展》汇集了微分几何、几何分析、微分方程等方面一些综述性文章或最新进展研究成果，特别有益于几何、分析专业的硕士生、博士生以及刚刚开始从事这些领域研究的数学工作者，对在相关领域工作的其他数学工作者也很有参考价值。

书籍目录

Minimal Volume and Simplicial Volume of Visibility n—manifolds and All 3—manifblds 1 Introduction 2 Simplicial Volume and Bounded Cohomology 3 Visibility Manifolds and Gromov—hyperbolic Spaces 4 Gromov’S Simplicial Volume of Visibility n—manifoIds and Compact 3—manifolds Reference Rigidity Theorems for Lagrangian Submanifolds of Complex Space Forms with Conformal Maslov Form 1 Introduction 2 Preliminaries 3 Rigidity Theorems for Lagrangian Submanifolds References Method of Moving Planes in Integral Forms and Regularity Lifting 1 Introduction 2 Illustration of MMP in Integral Forms 3 Various Applications of MMP in Integral Forms 4 Regularity Lifting by Contracting Operators 5 Regularity Lifting by Combinations of Contracting and Shrinking Operators References The Ricci Curvature in Finsler Geometry 1 Definitions and Notations. 2 Ricci Curvature of Randers Metrics 3 Volume Comparison in Finsler Geometry 4 The Role of the Ricci Curvature in Projective Geometry RefeFences Specific Non—K（a）hler Hermitian Metrics on Compact Complex Manifolds 1 Introduction 2 Balanced Metrics Under the Conifold Transition 3 The Supersymmetric Solutions 4 The Form—type Calabi—Yau Equation 5 The Generalized Gauduchon Metrics RefeFences On the σ2—scalar Curvature and Its Applications 1 Introduction 2 σ2—Yamabe Problem 3 A Quotient Equation 4 Ellipticity of a Quotient Equation and a 3—dimensional Sphere Theorem 5 A Rough Classification of Metrics of Positive Scalar Curvature 6 An Almost Schur Theorem References Isometric Embedding of Surfaces in R3 1 Introduction 2 Local Isometric Embedding of Surfaces 3 Global Isometric Embedding of Surfaces References The Lagrangian Mean Curvature Flow Along the K（a）hler—Ricci Flow 1 Introduction 2 Lagrangian Property Is Preserved RefeFences On Magnetohydrodynamics with Partial Magnetic Dissipation near Equilibrium 1 Introduction 2 Reformulation of the Ideal MHD 3 Global Existence 4 Acknowledgement Refefences Hyperbolic Gradient Flow：Evolution of Graphs in Rn+1 1 Introduction 2 Hyperbolic G radient Flow for Graphs in Rn+1 3 The Evolution of Convex Hypersurfaces in Rn+1 4 The Evolution of Plane Curves 5 Conclusions and Open Problems References The Moser—Trudinger and Adams Inequalities and Elliptic and Subelliptic Equations with Nonlinearity of Exponential Growth 1 Introduction 2 The Moser—nudinger Inequalities 3 Adams Type Inequalities on High Order Sobolev Spaces 4 N—Laplacian on Bounded Domains in RN 5 Polyharmonic Equations on Bounded Domains in R2m 6 N—Laplacian Equations with Critical Exponential Growth in RN 7 Existence of Solutions to Polyharmonic Equations with Critical Exponential Growth in the Whole Space 8 Subcritical Exponential Growth in Bounded Domains 9 Equations of Q—sub—Laplacian Type with Critical Growth on Bounded Domains 10 Equations of Q—sub—Laplacian Type with Critical Growth in Hn 1 1 Examples of Nonlinear Terms Without the Ambrosetti—Rabinowitz Condition 12 Sharp Moser—Trudinger Inequality on the Heisenberg Group at the Critical Case and Multiplicity of Solutions 13 A New Approach to Sharp Adams Inequalities for Arbitrary Integers and Less Restrictive Norms References Navigation Problem and Randers Metrics 1 Historical Remarks and Definitions 2 Navigation Problem 3 Flag Curvature Decreasing Property of Navigation Problem 4 New Finsler Metrics with Constant（Scalar）Flag Curvature from Old 5 Geodesics of a Finsler Metric Via Navigation Problem 6 A New Characterization of Randers Norms 7 Compact Randers Metrics with Constant S—curvature 8 Classification Results for Randers Metrics 9 Construction of Randers Metrics with Isotropic S—curvature. References Lorentzian Isoparametric Hypersurfaces in the Lorentzian Sphere Sn+1，1 1 Introduction 2 Canonical Forms for Symmetric Transformation A in Lorentzian Space 3 Lorentzian Isoparametric Hypersurface of Type Ⅰ in Sn+1，1 4 Nonexistence of Lorentzian Isoparametric Hypersurface of Type Ⅳ in Sn+1，1 5 Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn+1，1 6 Totally Umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn+1，1 7 Semi—umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn+1，1 8 Non—umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn+1，1 References Asymptotically Hyperbolic Manifolds and Conformal Geometry 1 Definitions 2 Regularity and Rigidity for AH Manifold 3 Correspondences 4 Generalized Yamabe Problems References A Characterization of Randers Metrics of Scalar Flag Curvature 1 Introduction 2 Preliminaries 3 A Formula for the Weyl Curvature 4 Proof of Theorem 1.1 5 Weak Einstein Metrics 6 Proof of Theorem 1.2 References Prescribed Weingarten Curvature Equations Geometry Problems Related with Quasi—local Mass in General Relativity Concerning the L4 Norms of Typical Eigenfunctions on Compact Surfaces Analysis on Riemannian Manifolds with Non—convex Boundary The Unity of p—harmonic Geometry Hermitian Harmonic Maps Between Almost Hermitian Manifolds A Global Mean Value Inequality for Plurisubharmonic Functions on a Compact K（a）hler Manifold A List of Publications by Professor Zhengguo Bai A List of Publications by Professor Yibing Shen A List of Graduate Students of Professors Bai and Shen

章节摘录

版权页：   插图：   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)

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