本书汇集了微分几何、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。
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- Front Matter
- Minimal Volume and Simplicial Volume of Visibility n-manifolds and All 3-manifolds Jianguo Cao, Xiaoyang Chen
- 1 Introduction
- 2 Simplicial Volume and Bounded Cohomology
- 3 Visibility Manifolds and Gromov-hyperbolic Spaces
- 4 Gromov’s Simplicial Volume of Visibility n-manifolds and Compact 3-manifolds
- References
- Rigidity Theorems for Lagrangian Submanifolds of Complex Space Forms with Conformal Maslov Form Xiaoli Chao, Yuxin Dong
- 1 Introduction
- 2 Preliminaries
- 3 Rigidity Theorems for Lagrangian Submanifolds
- References
- Method of Moving Planes in Integral Forms and Regularity LiftingWenxiong Chen, Congming Li
- 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 Xinyue Cheng
- 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
- References
- Specific Non-K¨ahler Hermitian Metrics on Compact Complex Manifolds Jixiang Fu
- 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
- References
- On the _2-scalar Curvature and Its Applications Yuxin Ge, Guofang Wang
- 1 Introduction
- 2 _2-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 Qing Han
- 1 Introduction
- 2 Local Isometric Embedding of Surfaces
- 3 Global Isometric Embedding of Surfaces
- References
- The Lagrangian Mean Curvature Flow Along the K¨ahler-Ricci Flow Xiaoli Han, Jiayu Li
- 1 Introduction
- 2 Lagrangian Property Is Preserved
- References
- On Magnetohydrodynamics with Partial Magnetic Dissipation near Equilibrium Xianpeng Hu, Zhen Lei, Fang-Hua Lin
- 1 Introduction
- 2 Reformulation of the Ideal MHD
- 3 Global Existence
- 4 Acknowledgement
- References
- Hyperbolic Gradient Flow: Evolution of Graphs in Rn+1 De-Xing Kong, Kefeng Liu
- 1 Introduction
- 2 Hyperbolic Gradient 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 Nguyen Lam, Guozhen Lu
- 1 Introduction
- 2 The Moser-Trudinger 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
- 11 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 Meng Li, Xiaohuan Mo, Ni Yu
- 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+11 Zhen-Qi Li
- 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+11
- 6 Totally Umbilical Lorentzian Isoparametric Hypersurface of Type II in Sn+1
- 7 Semi-umbilical Lorentzian Isoparametric Hypersurface of Type II in Sn+1
- 8 Non-umbilical Lorentzian Isoparametric Hypersurface of Type II in Sn+1
- References
- Asymptotically Hyperbolic Manifolds and Conformal Geometry Jie Qing
- 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 Zhongmin Shen, Gulcin Civi
- 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 Weimin Sheng, Neil S Trudinger, Xu-Jia Wang
- 1 Introduction
- 2 The Uniform Estimate
- 3 The Gradient Estimates
- 4 The Second Order Derivative Estimate
- 5 The Dirichlet Problem
- 6 Closed Convex Hypersurfaces
- References
- Geometry Problems Related with Quasi-local Mass in General Relativity Yuguang Shi
- 1 Introduction
- 2 Quasi-local Mass on Domains with Convex Boundaries
- 3 Quasi-local Mass on Domains with Non Convex Boundary
- References
- Concerning the L4 Norms of Typical Eigenfunctions on Compact Surfaces Christopher D Sogge, Steve Zelditch
- 1 Introduction
- 2 L4 Norms of Generic Eigenfunctions
- 3 Average L4 Norms of Spherical Harmonics
- 4 L4 Norms of Orthonormal Bases of Spherical Harmonics
- 5 Random Orthonormal Bases of Spherical Harmonics
- 6 Other Orthonormal Bases
- References
- Analysis on Riemannian Manifolds with Non-convex Boundary Feng-Yu Wang
- 1 Introduction
- 2 Semigroup Properties for Curvature and Second
- Fundamental Form
- 3 Dimension-free Harnack Inequalities
- 4 Functional Inequalities
- References
- The Unity of p-harmonic Geometry Shihshu Walter Wei
- 1 Introduction
- 2 A Perspective of p-harmonic Maps
- 3 p-Harmonic Maps and Cohomology Groups
- 4 Unified Notions of F-harmonic Maps
- 5 Unified Notions of F-Yang-Mills Fields
- 6 Vector Bundle Valued p-forms on Riemannian Manifolds:A Higher Level of Generalization
- 7 Manifolds with the Global Doubling Property in p-harmonic Geometry
- 8 Generalized 1-harmonic Equations and Geometric Flows
- 9 The Mean Curvature Flow
- 10 Generalized Constant Mean Curvature Type Equations for Differential Forms
- 11 Geometric Inequalities on Manifolds
- 12 Harmonic Maps in Finsler Geometry
- References
- Hermitian Harmonic Maps Between Almost Hermitian Manifolds Xi Zhang
- 1 Introduction
- 2 Preliminary Results
- 3 Proof of Theorem 1.1
- References
- A Global Mean Value Inequality for Plurisubharmonic Functions on a Compact K¨ahler Manifold Xiaohua Zhu
- 1 Introduction
- 2 Relative Capacity and L1-estimate
- 3 Proof of Theorem 1.1
- 4 A Discussion on Theorem 1.1
- References
- 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
- 版权
