Mathematical relativity and asymptotically hyperbolic manifolds

如果说有两种几何大家都很熟悉的话，那就是欧几里得的平面几何和球面的圆几何。第三种，“马鞍形”双曲几何，不太熟悉，但由艺术家MC Bertler的工作推广。这三个模型代表恒定曲率的几何形状，并表现出最大的对称性，在每个方向和每个点上看起来都相同。渐近双曲（AH）流形，以及它们的近亲渐近反德西特（AAdS）时空，对称性较低。它们的中间可能是凹凸不平和不规则的，但在离这个中心区域很远的地方，它们越来越像双曲几何。它们在过去30年的现代物理学中起着重要的作用。它们是黑洞热力学的天然竞技场，并出现在AdS/CFT对应中，将这些几何与量子共形场论联系起来，并体现了“全息”，即一个区域内的物理学由该区域边界上的其他物理学编码的推测性概念。时空的几何学是由爱因斯坦的广义相对论支配的。爱因斯坦的方程包含了丰富多样的AH和AAdS几何。有些没有物质，也没有黑洞，但却有质量--事实上，是负质量！并且具有不寻常的形状或拓扑结构。这项研究计划旨在研究这些迷人的几何形状。在其他问题中，它会问什么时候质量可以是负的，它能有多负，以及这与拓扑结构有什么关系？它还将询问是否在由其他几何方程（即四阶偏微分方程）支配的几何中看到类似的结构。几何体可以变形，使其更平滑、更对称。这种变形是数学家和物理学家的重要工具。这个提议的一部分涉及研究被里奇流变形的AH几何，里奇流是最近用来证明庞加莱猜想的一种方法。该提案旨在确定由Ricci流变形的AH几何形状的详细演变。这个提议的另一部分涉及爱因斯坦方程的推广，其中Ricci曲率张量被更一般的对象Bakry-Émery-Ricci张量取代。这里的问题是，爱因斯坦理论的数学结构是否真的是由该理论产生的几何所独有的，还是也为其他更一般的几何所共有。简而言之，该提案寻求物理学中重要问题的答案，无论是在AdS/CFT对应还是在广义相对论中，都利用了渐近双曲流形，几何流和密度流形的最新数学进展，具有Bakry-Émery-Ricci下界。

If there are two kinds of geometry generally familiar, they are the flat geometry of Euclid and the round geometry of spheres. A third kind, "saddle shaped'' hyperbolic geometry, is less familiar but was popularized by the work of the artist MC Escher. These three models represent the constant curvature geometries and exhibit maximal symmetry, looking the same in every direction and at every point. Asymptotically hyperbolic (AH) manifolds, and their close cousins the asymptotically anti-de Sitter (AAdS) spacetimes, as less symmetrical. They may be bumpy and irregular in the middle but resemble hyperbolic geometry more and more at large distances from this central region. They play an important role in the modern physics of the last 30 years. They are a natural arena for black hole thermodynamics, and appear in the AdS/CFT correspondence, which relates these geometries to quantum conformal field theory and manifests "holography'', the speculative notion that physics within a region is encoded by other physics on the boundary of that region.******The geometry of spacetime is governed by Einstein's general relativity. Einstein's equations admit a rich variety of AH and AAdS geometries. Some contain no matter and no black holes, yet have mass---indeed, negative mass!---and have unusual shape or topology. This research proposal seeks to study these fascinating geometries. Among other questions, it will ask when can the mass be negative, how negative can it be, and how is this related to the topology? It will also ask whether similar structure is seen in geometries governed by other geometric equations, namely fourth order partial differential equations.******Geometries can be deformed to be made smoother and more symmetrical. Such deformations are important tools for mathematicians and physicists. Part of this proposal concerns the study of AH geometries that are deformed by Ricci flow, a method used recently to prove the Poincaré conjecture. The proposal seeks to determine the detailed evolution of AH geometries deformed by Ricci flow.******Another part of this proposal involves a generalization of Einstein's equations in which the Ricci curvature tensor is replaced by a more general object, the Bakry-Émery-Ricci tensor. Here the question is whether the mathematical structure of Einstein's theory is really exclusive to the geometries that arise from that theory or is shared by other more general geometries as well.******In short, the proposal seeks answers to important questions in physics, both in the AdS/CFT correspondence and in general relativity, by leveraging recent mathematical advances in asymptotically hyperbolic manifolds, geometric flows, and manifolds-with-density with a Bakry-Émery-Ricci lower bound.