Hubert L.Bray、William P.Minicozzi所著的《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。

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。Hubert L.Bray、William P.Minicozzi所著的《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。

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 Birkhoff: A Closed Geodesic on a Two Sphere

 5 Curve Shortening Flow

 6 Minimal Surfaces

 7 Classification of Embedded Minimal Surfaces

 8 Mean Curvature Flow

 9 Width and mean curvature flow

 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 Differentiability Theorem of Gromov-Schoen

 Georgios Daskalopoulos, Chikako Mese

 1 Introduction

 2 Definitions

 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, Rare 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 Definitions 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 Kaihler Case

 4 Quaternionic Kahler 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 Hun 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

 Andre 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 1.1 and 1.2

 6 Proof of Theorem 1.3

 A Geometry of Quotients

 B Orbifolds

 References

Deformation of Kaihler-Einstein Metrics

 Xiaofeng Sun, Shing-Tung Yau

 1 Introduction

 2 Complex Structures of Kahler-Einstein Manifolds

 3 Deformation of Kahler-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 Diffeomorphisms Between Complete Noncompact Surfaces

 Tom Y. H. Wan

 1 Introduction

 2 Harmonic Map Theory of Universal Teichmiiller Space

 3 Asymptotic Behavior of Open Harmonic Embedding From the Complex Plane Into Hyperbolic Plane

 References

Compactifications of Complete Riemannian Manifolds and Their Applications

 Xiaodong Wang

 1 Introduction

 2 The Geometric Compactification

 3 The Martin Compactification

 4 The Busemann Boundary

 5 A Comparison Theorem

 References

Some Aspects of Weil-Petersson Geometry of Teichmuller 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 Teichmuller Space

 7 Embeddings of the Coxeter Complex into UT

 8 Summary and Open Problems

 References