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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 Classiˉcation 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 Deˉnitions
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 Deˉnitions 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 K?ahler Case
4 Quaternionic K?ahler 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 K?ahler-Einstein Metrics Xiaofeng Sun, Shing-Tung Yau
1 Introduction
2 Complex Structures of K?ahler-Einstein Manifolds
3 Deformation of K?ahler-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 Teichm?uller Space
3 Asymptotic Behavior of Open Harmonic Embedding From
the Complex Plane Into Hyperbolic Plane
References
Compactiˉcations of Complete Riemannian Manifolds and Their Applications Xiaodong Wang
1 Introduction
2 The Geometric Compactiˉcation
3 The Martin Compactiˉcation
4 The Busemann Boundary
5 A Comparison Theorem
References
Some Aspects of Weil-Petersson Geometry of Teichm?uller 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 Teichm?uller Space
7 Embeddings of the Coxeter Complex into UT
8 Summary and Open Problems
References
