自从爱因斯坦提出广义相对论以来，微分几何就与广义相对论密不可分。微分几何和几何分析为学习广义相对论提供方法以及正确的框架，而广义相对论激发富有挑 战性的各种问题。《几何分析与相对论》包含23篇几何分析和广义相对论各领域的综述性文章，作者均为该领域的知名专家。几何分析方面的内容包括 Yamabe问题、平均曲率流、极小曲面、调和映照、Ricci流、胶合与分裂结构、函数论、流形的塌陷、Kahler-Einstein度量、完备流形 的渐近几何以及Teichmuller空间几何等。广义相对论方面的内容包括正质量定理、Penrose不等式、标量曲率及Einstein约束方程、准 局域质量泛函、高维黑洞拓扑、渐近双曲流形的正质量定理等。《几何分析与相对论》可供几何分析或相对论领域的研究人员和研究生参考。
- 目录
- 前辅文
- On the Positive Mass, Penrose, and ZAS Inequalities in General Dimension Hubert L Bray
- 1 Dedication
- 2 Introduction
- 3 A Trio of Inequalities
- References
- Recent Progress on the Yamabe Problem Simon Brendle, Fernando C Marques
- 1 The Yamabe Problem
- 2 The Compactness Conjecture
- 3 Non-compactness Results in Dimension n ¸ 25
- 4 A Compactness Result in Dimension n • 24
- 5 The Parabolic Yamabe Flow
- References
- Some Recent Progress on Mean Curvature Flow for Entire Lagrangian Graphs Jingyi Chen
- 1 Introduction
- 2 Longtime Existence With Lipschitz Continuous Initial Data
- 3 Uniqueness and Viscosity Solutions
- 4 Self-similar Solutions
- References
- Radial Viewpoint on Minimal Surfaces Jaigyoung Choe
- 1 Introduction
- 2 Cone
- 3 Horizon
- 4 Non-Euclidean Space
- 5 Ray preserving Metric
- 6 Varying Curvature
- 7 Embeddedness
- References
- Minimal Surfaces and Mean Curvature Flow Tobias H Colding, William P Minicozzi II
- 1 Introduction
- 2 Harmonic Functions and the Heat Equation
- 3 Energy of a Curve
- 4 Birkho®: A Closed Geodesic on a Two Sphere
- 5 Curve Shortening Flow
- 6 Minimal Surfaces
- 7 Classi¯cation of Embedded Minimal Surfaces
- 8 Mean Curvature Flow
- 9 Width and mean curvature °ow
- 10 Singularities for MCF
- 11 Smooth Compactness Theorem for Self-shrinkers
- 12 The Entropy
- 13 An Application
- 14 Non-compact self-shrinkers
- References
- Scalar Curvature and the Einstein Constraint Equations Justin Corvino, Daniel Pollack
- 1 Introduction
- 2 The Constraint Equations
- 3 A Tour of Asymptotically Flat Solutions
- 4 The Conformal Method
- 5 Gluing Constructions
- References
- On the Intrinsic Di®erentiability Theorem of Gromov-Schoen Georgios Daskalopoulos, Chikako Mese
- 1 Introduction
- 2 De¯nitions
- 3 Main Theorem
- References
- Minimal Surface Techniques in Riemannian Geometry Ailana Fraser
- 1 Introduction
- 2 Brief Overview of Some Geodesic Methods
- 3 Existence of Minimal Surfaces
- 4 Second Variation Theory for Minimal Surfaces and Applications
- References
- Stability and Rigidity of Extremal Surfaces in Riemannian Geometry and General Relativity Gregory J Galloway
- 1 Minimal Hypersurfaces in Manifolds of Nonnegative
- Scalar Curvature
- 2 Marginally Outer Trapped Surfaces
- 3 Positivity of Mass for Asymptotically Hyperbolic Manifolds
- References
- Convex Hypersurfaces of Constant Curvature in Hyperbolic Space Bo Guan, Joel Spruck
- 1 Introduction
- 2 Formulas on Hypersurfaces
- 3 The Asymptotic Angle Maximum Principle and
- Gradient Estimates
- 4 Curvature Estimates
- 5 Uniqueness and Foliations
- References
- Ricci Flow in Two Dimensions James Isenberg, Rafe Mazzeo, Natasa Sesum
- 1 Introduction
- 2 General Considerations
- 3 Compact Surfaces
- 4 Open Surfaces
- 5 Flows on Incomplete Surfaces
- References
- Doubling and Desingularization Constructions for Minimal Surfaces Nikolaos Kapouleas
- 1 Introduction
- 2 Doubling Constructions
- 3 Desingularization Constructions
- 4 Minimal Surfaces in the Round Three-Sphere
- 5 The Building Blocks for the Desingularization Construction
- 6 An Initial Surface for the Desingularization Construction
- 7 The Family of Initial Surfaces for the
- Desingularization Construction
- 8 Main Estimates and Outline of the Proof
- References
- The Metric Properties of Lagrangians Yng-Ing Lee
- 1 Introduction
- 2 A Short Survey
- 3 De¯nitions and Properties
- 4 Singularities and Geometric Measure Theory
- 5 Gluing and Singular Perturbation
- References
- Structure of Complete Manifolds with Positive Spectrum Peter Li
- 1 Introduction
- 2 Riemannian Case
- 3 KÄahler Case
- 4 Quaternionic KÄahler Manifolds, Cayley Manifolds, and Locally
- Symmetric Spaces
- 5 Manifolds of Finite Volume
- 6 Further Generalizations
- References
- Topology of Sobolev Mappings and Associated Variational Problems Fang Hua Lin
- Introduction
- 1 Analytical and Topological Properties of Sobolev Maps
- 2 Singularity of Energy Minimizing Maps
- 3 Limits of Singular Sets of p-Energy Minimizing Maps
- References
- A Survey of Research on Boundary Behavior of Compact Manifolds via the Positive Mass Theorem Pengzi Miao
- 1 Introduction
- 2 Statement of the Positive Mass Theorem
- 3 On compact Manifolds with Nonnegative Scalar Curvature
- 4 On Compact Manifolds with Negative Scalar Curvature
- References
- Recent Progress on Singularities of Lagrangian Mean Curvature Flow Andr¶e Neves
- 1 Introduction
- 2 Preliminaries
- 3 Basic Techniques
- 4 Applications I: Blow-ups
- 5 Applications II: Self-Expanders
- 6 Application III: Stability of Singularities
- 7 Open Questions
- References
- Geometric Structures of Collapsing Riemannian Manifolds I Aaron Naber, Gang Tian
- 1 Introduction
- 2 Structure of Collapsed Spaces
- 3 Geometry of Toric Quotients
- 4 Geometry of Toric Quotients II
- 5 Proof of Theorems 11 and 12
- 6 Proof of Theorem 13
- A Geometry of Quotients
- B Orbifolds
- References
- Deformation of KÄahler-Einstein Metrics Xiaofeng Sun, Shing-Tung Yau
- 1 Introduction
- 2 Complex Structures of KÄahler-Einstein Manifolds
- 3 Deformation of KÄahler-Einstein Metrics
- 4 Local Trivialization of Polarization Bundles and Deformation of Sections
- 5 Curvature of L2 Metrics on Direct Image Sheaves
- 6 Appendix
- References
- Reverse Bubbling in Geometric Flows
- Peter M Topping
- 1 Introduction
- 2 The Harmonic map Flow
- 3 Ricci Flow
- 4 Addendum | Mean Curvature Flow
- References
- Review on Harmonic Di®eomorphisms Between Complete Noncompact Surfaces Tom Y H Wan
- 1 Introduction
- 2 Harmonic Map Theory of Universal TeichmÄuller Space
- 3 Asymptotic Behavior of Open Harmonic Embedding From
- the Complex Plane Into Hyperbolic Plane
- References
- Compacti¯cations of Complete Riemannian Manifolds and Their Applications Xiaodong Wang
- 1 Introduction
- 2 The Geometric Compacti¯cation
- 3 The Martin Compacti¯cation
- 4 The Busemann Boundary
- 5 A Comparison Theorem
- References
- Some Aspects of Weil-Petersson Geometry of TeichmÄuller Spaces Sumio Yamada
- 1 Introduction
- 2 Harmonic Maps into T and an Application
- 3 Finite Rank Properties of T
- 4 Coxeter-Tits Construction
- 5 Weil-Petersson Geodesic Completeness
- 6 Weil-Petersson Geometry of the Universal TeichmÄuller Space
- 7 Embeddings of the Coxeter Complex into UT
- 8 Summary and Open Problems
- References
