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。本书包含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。本书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 generaldimension
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 lagrangiangraphs
on the positive mass, penrose, and zas inequalities in generaldimension
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 lagrangiangraphs
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
a ilana fraser
1 introduction
2 brief overview of some geodesic methods
3 existence of minimal surfaces
4 second variation theory for minimal surfaces andapplications
references
stability and rigidity of extremal surfaces in riemannian
geometry and general relativity
gregory j galloway
1 minimal hypersurfaces in manifolds of nonnegative scalarcurvature
2 marginally outer trapped surfaces
3 positivity of mass for asymptotically hyperbolic manifolds
references
convex hypersurfaces of constant curvature in hyperbolicspace
bo guan, joel spruck
1 introduction
2 formulas on hypersurfaces
3 the asymptotic angle maximum principle and gradientestimates
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 minimalsurfaces
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 desingularizationconstruction
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 kahler case
4 quaternionic kahler manifolds, cayley manifolds, and locallysymmetric 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
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 11 and 12
6 proof of theorem 13
a geometry of quotients
b orbifolds
references
deformation of kahler-einstein metrics
xiaofeng sun, $hing-tung yam
1 introduction
2 complex structures of kahler-einstein manifolds
3 deformation of kahler-einstein metrics
4 local trivialization of polarization bundles and deformation ofsections
5 curvature of l2 metrics on direct hnage 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 noncompactsurfaces
tom y h wan
1 introduction
2 harmonic map theory of universal teichmiiller space
3 asymptotic behavior of open harmonic embedding from the complexplane into hyperbolic plane
references
compactifications of complete riemannian manifolds and theirapplications
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 teichmiillerspaces
sumio yamada
1 introduction
2 harmonic maps into t and an application
3 finite rank properties of
4 coxeter-tits construction
5 weil-petersson geodesic completeness
6 weil-petersson geometry of the universal teichmfiller space
7 embeddings of the coxeter complex into ut
8 summary and open problems
references
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