幾何分析與相對論（簡體書）

自從愛因斯坦提出廣義相對論以來，微分幾何就與廣義相對論密不可分。微分幾何和幾何分析為學習廣義相對論提供方法以及正確的框架，而廣義相對論激發富有挑戰性的各種問題。《幾何分析與相對論》包含23篇幾何分析和廣義相對論各領域的綜述性文章，作者均為該領域的知名專家。幾何分析方面的內容包括Yamabe問題、平均曲率流、極小曲面、調和映照、Ricci流、膠合與分裂結構、函數論、流形的塌陷、Kahler-Einstein度量、完備流形的漸近幾何以及Teichmuller空間幾何等。廣義相對論方面的內容包括正質量定理、Penrose不等式、標量曲率及Einstein約束方程、準局域質量泛函、高維黑洞拓撲、漸近雙曲流形的正質量定理等。《幾何分析與相對論》可供幾何分析或相對論領域的研究人員和研究生參考。

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of lagrangiansyng-ing lee1 introduction2 a short survey3 definitions and properties4 singularities and geometric measure theory5 gluing and singular perturbationreferencesstructure of complete manifolds with positive spectrumpeter li1 introduction2 riemannian case3 kahler case4 quaternionic kahler manifolds, cayley manifolds, and locallysymmetric spaces5 manifolds of finite volume6 further generalizationsreferencestopology of sobolev mappings and associated variationalproblemsfang hua linintroduction1 analytical and topological properties of sobolev maps2 singularity of energy minimizing maps3 limits of singular sets of p-energy minimizing mapsreferencesa survey of research on boundary behavior of compactmanifolds via the positive mass theorempengzi miao1 introduction2 statement of the positive mass theorem3 on compact manifolds with nonnegative scalar curvature4 on compact manifolds with negative scalar curvaturereferencesrecent progress on singularities of lagrangian meancurvature flowandre neves1 introduction2 preliminaries3 basic techniques4 applications i: blow-ups5 applications ii: self-expanders6 application iii: stability of singularities7 open questionsreferencesgeometric structures of collapsing riemannian manifolds iaaron naber, gang tian1 introduction2 structure of collapsed spaces3 geometry of toric quotients4 geometry of toric quotients ii5 proof of theorems 11 and 126 proof of theorem 13a geometry of quotientsb orbifoldsreferencesdeformation of kahler-einstein metricsxiaofeng sun, $hing-tung yam1 introduction2 complex structures of kahler-einstein manifolds3 deformation of kahler-einstein metrics4 local trivialization of polarization bundles and deformation ofsections5 curvature of l2 metrics on direct hnage sheaves6 appendixreferencesreverse bubbling in geometric flowspeter m topping1 introduction2 the harmonic map flow3 ricci flow4 addendum -- mean curvature flowreferencesreview on harmonic diffeomorphisms between complete noncompactsurfacestom y h wan1 introduction2 harmonic map theory of universal teichmiiller space3 asymptotic behavior of open harmonic embedding from the complexplane into hyperbolic planereferencescompactifications of complete riemannian manifolds and theirapplicationsxiaodong wang1 introduction2 the geometric compactification3 the martin compactification4 the busemann boundary5 a comparison theoremreferencessome aspects of weil-petersson geometry of teichmiillerspacessumio yamada1 introduction2 harmonic maps into t and an application3 finite rank properties of4 coxeter-tits construction5 weil-petersson geodesic completeness6 weil-petersson geometry of the universal teichmfiller space7 embeddings of the coxeter complex into ut8 summary and open problemsreferences