New geometrical perspectives in general relativity

在过去的几十年里，两股驱动力推动了广义相对论的研究。一个是来自“大科学”的数据的冲击，例如宇宙微波背景观测和LIGO引力波观测。另一个是几何分析的爆炸性增长，几何分析是过去半个世纪数学中最成功的领域之一。这项提议旨在利用几何分析工具，并将其应用于广义相对论中的问题。一个这样的问题是黑洞视界可能的拓扑结构。这反过来又是黑洞唯一性问题的一部分。虽然这些问题大多在4个时空维度上得到了很好的理解，但在我们从所有维度和非零宇宙常数理解它们之前，这个理论是不完整的。这里可以利用的两个工具是静态真空爱因斯坦度规方程和近视界几何方程。这两类方程都是贝克里-埃默里·里奇的“准爱因斯坦”方程，在里奇流动理论中很重要的里奇孤子方程也是如此。我建议从一个统一的观点来研究这些方程，这将使我能够借鉴一种贝克里-埃默里·利玛希方程的技巧，并利用它来研究另一种方程。这一策略已经产生了早期的结果，包括一个令人惊讶的应用于解释CMB数据：Gloway，Khuri和我已经能够证明，无论何时宇宙关闭，允许的宇宙拓扑都可以受到限制，即使宇宙没有达到封闭密度(这是封闭宇宙的充分条件，但不是必要条件)。该提议的第二个相关部分是研究渐近双曲时空，特别是静态真空时空。一个令人烦恼的问题是Horowitz和Myers的一个古老的猜想，即如果宇宙常数为负，那么对于共形无穷大是环面而不是球面的孤立系统，存在一种正质量定理(允许一些负质量，但有下限)。最近的进展表明，这个猜想在某些非常特殊的情况下成立，我建议推广这些结果。但当去掉特殊的对称性假设后，这一猜想的主要检验将到来。特别是，当共形无穷大是一个环面，但时空包含一组具有拓扑学性质的球形黑洞，其相互作用势能可能贡献负质量能量时，这个猜想是否成立？最后，我建议研究曲线缩短流(CSF)及其相关的外在几何流的孤子解。如果曲线以其曲率给定的速度在法线方向上变形，则曲线通过曲线缩短而演化。孤子是在流动下自相似演化的曲线。但人们对弯曲流形中的脑脊液孤子知之甚少。这将是本科生研究项目的丰富来源。我希望将这一结果应用于宇宙学中的宇宙弦进化。

Two driving forces have fueled general relativity research in the past few decades. One is the onslaught of data from ``big science'', such as the cosmic microwave background (CMB) observations and LIGO gravitational wave observations. The other is the explosive growth of geometric analysis, one of the most successful fields in mathematics in the last half-century. This proposal aims to leverage geometric analytical tools and to apply them to problems in general relativity. One such problem is that of the possible topology of black hole horizons. This in turn is part of the black hole uniqueness problem. While these problems are mostly well understood in 4 spacetime dimensions, the theory will not be complete until we understand them in all dimensions and for nonzero cosmological constant. Two tools to exploit here are the equations of static vacuum Einstein metrics and the near horizon geometry equations. These are both types of Bakry-Émery Ricci "quasi-Einstein" equations, as are the Ricci soliton equations which are important in the theory of Ricci flow. I propose to study these equations from a unified viewpoint, which will enable me to borrow techniques from one kind of Bakry-Émery Ricci equation and exploit it to study another. This strategy has already yielded early results, including a surprising application to interpretation of CMB data: Galloway, Khuri, and I have been able to show that the allowed topologies for the Universe can be constrained whenever the Universe is closed, even if the Universe does not achieve closure density (which is a sufficient but not necessary condition for a closed Universe). A second, related part of the proposal is the study of asymptotically hyperbolic spacetimes, especially static vacuum ones. A vexing problem is an old conjecture of Horowitz and Myers that there exists a kind of positive mass theorem (allowing some negative mass, but with lower bound) for isolated systems whose conformal infinity is that of a torus rather than a sphere, if the cosmological constant is negative. Recent progress has shown that the conjecture holds in some quite special circumstances, and I propose to extend those results. But the main test of the conjecture will come when special symmetry assumptions are removed. In particular, can the conjecture hold when conformal infinity is a torus but the spacetime contains an array of topologically spherical black holes whose interaction potential energy may contribute negative mass-energy? Finally, I propose to study soliton solutions of the curve shortening flow (CSF) and related extrinsic geometric flows. A curve evolves by curve shortening if it deforms in the normal direction with speed given by its curvature. Solitons are curves that evolve self-similarly under the flow. But little is known of CSF solitons in curved manifolds. This will be a rich source of research projects for undergraduates. I expect to apply the results to cosmic string evolution in cosmology.