Differential geometric problems in mathematical relativity

主要研究方向：(I)广义相对论中初始数据集的几何和拓扑，以及(Ii)时空刚性的结果。时空M中的初始数据集由光滑的类空超曲面V、其诱导度规h和第二基本形式K组成。V中的一个闭合的双侧超曲面(MOTS)，其出射的光线具有零发散。MOTS的存在标志着一种引力极端的情况：一般而言，人们预计奇点的发展和黑洞的形成。莫茨斯是在更纯粹的数学背景下出现的，当时舒恩和尤的工作涉及到张氏方程解的存在性，这与他们对质量正性的证明有关。在时间对称(K=0)的初始数据集中，MOTS仅仅是一个(双边)极小曲面。多年来，极小曲面理论的一个重要主题就是利用极小曲面来研究黎曼流形的拓扑。类似地，在拓扑审查结果(在时间结果上是全局的)的激励下，PI及其合作者M.Eichmair和D.Pollack最近研究了三维渐近平坦初始数据集的拓扑与Motss的发生之间的关系。特别地，利用几何化的一个结果和最近关于MoTSS的存在性结果，他们证明了非平凡的拓扑蕴含着(浸入的)MoTss的存在。这可以解释为Meek-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建议进行研究，特别是在具有内视界(MOTS边界)的高维初始数据流形的背景下，这些高维初始数据流形满足物理上的自然曲率条件--主导能量条件。这种方法利用了张氏方程和Motss方程的解之间的联系。与上述工作相关的各种刚性问题也将被讨论，PI以前与R.Schoen在黑洞拓扑上的工作，以及PI最近与Carlos Vega关于时空刚性问题的工作，该问题最初是由Yau提出的，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性质的。引力场和其他场、黑洞和相关物体可以用几何方法来描述和分析。更广泛地说，这个项目涉及从几何角度研究当前科学兴趣的某些重力特征，利用黎曼几何、空间数学理论和洛伦兹几何工具，洛伦兹几何是时空数学理论。这些理论为研究时空宇宙的三个基本方面之间的关系提供了一种方法：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)。

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).