Metric Differential Geometry and Mathematical Gravity

AbstractAward：DMS-0708048首席研究员：Gregory J. Galloway该提案涉及时空几何和广义相对论界面的研究项目。 特别是，边缘捕获表面的性质，这在黑洞理论中发挥了重要作用，将被调查。 虽然物理上很好的动机，很少严格的结果，边缘捕获表面，和相关的对象，如动态地平线，一直到最近才知道。 这种情况已经改变，现在有严格的结果，关于存在性，唯一性，和其他问题。本研究的部分动机是实现边缘捕获曲面与黎曼流形中的极小曲面之间更深入的联系。 该项目将对一些与边际捕获表面有关的主题进行研究，例如黑洞的拓扑结构，最外层边际捕获表面的刚性和规则性，以及动力学视界的各个方面。弦论增加了人们对高维引力的兴趣，特别是最近有大量关于高维时空中黑洞的研究。Schoen和PI最近将Hawking关于黑洞拓扑的非经典结果推广到了高维空间。本提案的一部分，涉及联合工作与Schoen，关注的努力，以获得一个importantstrengthening这一结果，这将关闭一个缺口，在有关允许的拓扑结构的结论。这个循环的想法也应该是有用的，在获得较低的熵边界为高维稳态黑洞的渐近AdS设置，扩展已知的结果在四维。PI还将与Andersson和Cai一起进行一个关于渐近双曲流形质量的正性的持续项目的研究，以及关于弦理论经典稳定性的Penrose问题的研究。引力场和其他场，黑洞和相关的天体，都可以用几何方法来描述和分析。更一般地说，这个项目是从几何学的角度，利用黎曼几何（一种空间的数学理论）和洛伦兹几何（一种时空的数学理论）的工具，研究当前科学感兴趣的引力的某些特征。这些理论提供了一种研究时空宇宙的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）和混沌结构结构（即，the large大scale规模behavior行为of light光rays射线and light光cone锥体）.

AbstractAward: DMS-0708048Principal Investigator: Gregory J. GallowayThis proposal is concerned with research projects at theinterface of spacetime geometry and general relativity. Inparticular, properties of marginally trapped surfaces, which haveplayed an important role in theory of black holes, will beinvestigated. Although physically well motivated, very fewrigorous results about marginally trapped surfaces, andassociated objects, such as dynamical horizons, had been knowntil recently. This situation has changed, and there are nowrigorous results concerning existence, uniqueness, and otherissues. Motivating this study in part is the realization of adeeper connection between marginally trapped surfaces inspacetime and minimal surfaces in Riemannian manifolds. The PIwill conduct research on a number of topics pertaining tomarginally trapped surfaces, such as the topology of black holes,rigidity and regularity of outermost marginally trapped surfaces,and aspects of dynamical horizons. String theory has increasedinterest in gravity in higher dimensions, and, in particular,there has been a great deal of recent research concerning blackholes in higher dimensional spacetimes. Schoen and the PIrecently obtained a generalization to higher dimensions of aclassical result of Hawking concerning the topology of blackholes. One part of the present proposal, involving joint workwith Schoen, concerns an effort to obtain an importantstrengthening of this result, which would close a gap in theconclusion concerning allowed topologies. This circle of ideasshould also be useful in obtaining lower entropy bounds forhigher dimensional stationary black holes in the asymptoticallyAdS setting that extend known results in four dimensions. The PIwill also conduct research on a continuing project with Anderssonand Cai concerning the positivity of mass for asymptoticallyhyperbolic manifolds, and research pertaining to a problem ofPenrose concerning the classical stability of string theory.Modern theories of gravity are geometrical in nature. Thegravitational field and other fields, black holes and relatedobjects, may be described and analyzed using geometricmethods. In more general terms, this project is concerned withthe study of certain features of gravity of current scientificinterest from this geometric point of view, utilizing the toolsof Riemannian geometry, a mathematical theory of space, andLorentzian geometry, a mathematical theory of spacetime. Thesetheories provide a method for studying the relationship amongthree fundamental aspects of the spacetime universe: curvature(i.e., the bending of space or spacetime), topology (i.e., theglobal shape and complexity of space or spacetime) and causalstructure structure (i.e., the large scale behavior of light raysand light cones).