Differential geometric problems in mathematical relativity

数学相对论中的微分几何问题

• 批准号: 1313724
• 负责人: Gregory Galloway
• 金额: $ 17.4万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313724&HistoricalAwards=false
• 关键词: Differential; geometric; problems; mathematical; relativity

AbstractAward: DMS 1313724, Principal Investigator: Gregory J. GallowayThe principal investigator proposes to continue his work on two basic research directions: (i) the geometry and topology of initial data sets in general relativity, and (ii) spacetime rigidity results.An initial data set in a spacetime M consists of a smooth spacelike hypersurface V, its induced metric h and second fundamental form K. A 'marginally outer trapped surface' (MOTS) in V is a closed two-sided hypersurface whose out-going light rays have vanishing divergence. The presence of a MOTS signals a gravitationally extreme situation: Generically one expects the development of singularities and the formation of a black hole. MOTSs arose in a more purely mathematical context in the work of Schoen and Yau concerning the existence of solutions to Jang's equation, in connection with their proof of positivity of mass. In a time-symmetric (K =0) initial data set a MOTS is simply a (2-sided) minimal surface. An important theme in minimal surface theory for many years has been the use of minimal surfaces to study the topology of Riemannian manifolds. In a similar vein, motivated by results on topological censorship (which are global in time results), the PI and collaborators, M. Eichmair and D. Pollack, have recently studied the relationship between the topology of 3-dimensional asymptotically flat initial data sets and the occurrence of MOTSs. In particular, using a consequence of geometrization and recent existence results for MOTSs, they have shown that non-trivial topology implies the existence of (immersed) MOTSs. This can be interpreted as a non-time symmetric version of results of Meeks- Simons-Yau. This work raises many interesting questions that the PI proposes to work on, particularly in the context of higher dimensional initial data manifolds with inner horizon (MOTS boundary) which satisfy the 'dominant energy condition', a physically natural curvature condition. The approach advocated makes use of the connection between solutions of Jang's equation and MOTSs. Various rigidity problems will also be addressed, in connection with the aforementioned work, the PI's previous work with R. Schoen on the topology of black holes and the PI's recent work with Carlos Vega concerning a spacetime rigidity problem originally posed by Yau, a concrete version of which is known as the Bartnik splitting conjecture.Modern theories of gravity are geometrical in nature. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In more general terms, this project is concerned with the study of certain features of gravity of current scientific interest from this geometric point of view, utilizing the tools of Riemannian geometry, a mathematical theory of space, and Lorentzian geometry, a mathematical theory of spacetime. These theories provide a method for studying the relationship among three fundamental aspects of the spacetime universe: curvature (i.e., the bending of space or spacetime), topology (i.e., the global shape and complexity of space or spacetime) and causal structure (i.e., the large scale behavior of light rays and light cones).

AbstractAward：DMS 1313724，首席研究员：Gregory J. Galloway首席研究员建议继续他在两个基本研究方向上的工作：（i）广义相对论中初始数据集的几何和拓扑，（ii）时空刚性结果。时空M中的初始数据集由光滑的类空超曲面V，它的诱导度量h和第二基本形式K组成。 V中的“边际外捕获面”（MOTS）是一个封闭的双边超曲面，其出射光线的发散度为零。MOTS的存在标志着一种引力极端情况：一般来说，人们期望奇点的发展和黑洞的形成。 MOTS出现在一个更纯粹的数学背景下的工作肖恩和丘有关的解决方案的存在张的方程，在连接与他们的证明积极的质量。在时间对称（K =0）的初始数据集中，MOTS只是一个（2边）极小曲面。 多年来，极小曲面理论中的一个重要主题是利用极小曲面来研究黎曼流形的拓扑。 在类似的脉络中，由拓扑审查的结果（这是全球性的时间结果）的动机，PI和合作者，M。Eichmair和D. Pollack等人最近研究了三维渐近平坦初始数据集的拓扑结构与MOTS发生之间的关系。 特别是，使用的结果几何化和最近存在的结果MOTS，他们已经表明，非平凡的拓扑结构意味着存在（浸入）MOTS。 这可以解释为Meeks-Simons-Yau结果的非时间对称版本。这项工作提出了许多有趣的问题，PI建议工作，特别是在高维初始数据流形与内视界（MOTS边界），满足“主导能量条件”，物理自然曲率条件的背景下。 主张的方法利用张方程的解和MOTS之间的联系。 各种刚性问题也将得到解决，与上述工作，PI的以前的工作与R。Schoen关于黑洞拓扑的研究，以及PI最近与卡洛斯·织女星（Vega）关于最初由Yau提出的时空刚性问题的研究，其具体版本被称为Bartnik分裂猜想。现代引力理论本质上是几何的。引力场和其他场，黑洞和相关的物体，可以用几何方法来描述和分析。更一般地说，该项目涉及从几何学的角度研究当前科学感兴趣的引力的某些特征，利用黎曼几何（空间的数学理论）和洛伦兹几何（时空的数学理论）的工具。这些理论提供了一种研究时空宇宙三个基本方面之间关系的方法：曲率（即，空间或时空的弯曲），拓扑（即，空间或时空的整体形状和复杂性）和因果结构（即，光线和光锥的大尺度行为）。