Metric Differential Geometry and Mathematical Gravity

项目编号：dms -0708048项目负责人：Gregory J. galloy本提案涉及时空几何与广义相对论界面的研究项目。特别是在黑洞理论中扮演重要角色的边缘俘获表面的特性，将被研究。尽管在物理上是有很好的动机的，但是关于边缘捕获表面和相关物体（如动态视界）的非常激烈的结果直到最近才为人所知。这种情况发生了变化，关于存在性、唯一性和其他问题没有令人信服的结果。推动这项研究的部分原因是实现了时空中边缘捕获曲面和黎曼流形中最小曲面之间更深层次的联系。该中心将研究与边缘捕获面有关的许多主题，如黑洞的拓扑结构、最外层边缘捕获面的刚性和规则性，以及动力学视界的各个方面。弦理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黑洞的研究。Schoen和pi最近将霍金关于黑洞拓扑的经典结果推广到更高的维度。本提案的一部分涉及与Schoen的联合工作，涉及努力获得对该结果的重要强化，这将缩小关于允许拓扑的结论中的空白。这个思想圈也应该有助于在渐近ad设置下获得高维平稳黑洞的低熵界，将已知结果扩展到四维。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。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，这个项目是利用黎曼几何（一种空间数学理论）和洛伦兹几何（一种时空数学理论）的工具，从这个几何角度研究当前科学感兴趣的重力的某些特征。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。拓扑结构（即空间或时空的整体形状和复杂性）和因果结构结构（即光线和光锥的大尺度行为）。

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).