Differential Geometric Problems in Mathematical Relativity

2016年，LIGO宣布探测到遥远黑洞产生的引力波，再次证实了爱因斯坦广义相对论的非凡力量。广义相对论是重力的几何理论；在这个理论中，引力的影响是由于宇宙的曲率。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。总的来说，这个项目是从几何角度研究当前科学感兴趣的重力的某些特征，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）。时空中的初始数据集由光滑的类空间超曲面、其诱导度规和第二基本形式组成。首席研究员（PI）将继续他关于视界渐近平坦初始数据集的研究。这样的初始数据集模拟了带有黑洞的孤立引力系统。在这种情况下，PI与各种合作者在初始数据水平上建立了“拓扑审查”的结果，用于三维和更高维度的初始数据集。这些结果建立了在自然物理假设下，对黑洞时空中外部通信空间域的几何和拓扑的限制。它们在很大程度上依赖于最近发展起来的边缘外俘获面理论（mots）。这些都是非常有趣的东西。一方面，它们在黑洞理论中起着重要的作用，另一方面，它们可以被视为黎曼几何中最小曲面的自然时空类似物。尽管MOTS不是一般的变分对象，但它们的性质与最小曲面非常相似。PI的工作和相关结果导致PI将研究许多有趣的开放问题，包括MOTS的拓扑问题和外部通信的空间域问题，以及关于MOTS在紧致和非紧致情况下的存在性和刚性现象的问题。拓扑问题将受益于对moss和所谓的浸入式moss之间关系的更深入理解。PI还将处理一些与“时间对称”（即完全测地线）初始数据集有关的问题。PI将在渐近欧几里得情况下的Schoen-Yau正质量证明的精神中，实施一种证明渐近双曲流形（AH）正质量定理的策略。这项研究有可能填补先前在AH病例中积极的大量证据的空白。在这个方向上的进一步目标是尝试用这种方法来获得关于AdS孤子的Horowitz-Myers猜想的证明。PI还将继续进行涉及光子球和时空刚性问题的正在进行的项目。

The remarkable detection, announced by LIGO in 2016, of gravitational waves generated by distant black holes has once again affirmed the extraordinary power of Einstein's General Theory of Relativity. General relativity is a geometric theory of gravity; in this theory the effects of gravity are due to the curvature of the universe. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In very 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). An initial data set in a spacetime consists of a smooth spacelike hypersurface, its induced metric and second fundamental form. The principal investigator (PI) will continue his investigations concerning asyptotically flat initial data sets with horizons. Such initial data sets model isolated gravitating systems with black holes. In this context the PI, with various collaborators, has established results on 'topological censorship' at the initial data level, for initial data sets in three and higher dimensions. These results establish, under natural physical assumptions, restrictions on the geometry and topology of the spatial domain of outer communication in black hole spacetimes. They rest in large measure on the rather recently developed theory of marginally outer trapped surfaces (MOTSs). These are objects of considerable interest. On the one hand they play an important role in the theory of black holes, and on the other, they may be viewed as natural spacetime analogues of minimal surfaces in Riemannian geometry. Although MOTS are not in general variational objects, they nevertheless turn out to have properties remarkably similar to minimal surfaces. The work of the PI, and related results, lead to many interesting open problems that the PI will investigate, including problems concerning the topology of MOTSs and the spatial domain of outer communication, as well as problems concerning MOTS existence and rigidity phenomena, both in the compact and noncompact case. Topological issues will benefit from a deeper understanding of the relationship between MOTSs and so-called immersed MOTSs. The PI will also work on some problems pertaining to `time-symmetric' (i.e., totally geodesic) initial data sets. The PI will implement a strategy for proving a positive mass theorem for asymptotically hyperbolic (AH) manifolds, in the spirit of the Schoen-Yau positive mass proof in the asymptotically Euclidean case. The research has the potential to close a gap in previous positive mass proofs in the AH case. A further aim in this direction would then be to try to use this method to obtain a proof of the Horowitz-Myers conjecture concerning the AdS soliton. The PI will also continue on-going projects involving photon spheres and spacetime rigidity problems.