Mathematical relativity and asymptotically hyperbolic manifolds

如果说有两种几何是人们普遍熟悉的，那就是欧几里得的平面几何和球面的圆形几何。第三种是“鞍形”双曲几何，不太为人所知，但在艺术家MC Escher的作品中得到了普及。这三个模型代表了常曲率几何，并展示了最大的对称性，在每个方向和每个点上看起来都是一样的。渐近双曲(AH)流形，以及它们的近亲渐近反De-Sitter(AADS)时空，因为它们不那么对称。它们的中间可能是凹凸不平和不规则的，但在远离这个中心区域的很远的地方，它们越来越像双曲几何。它们在过去30年的现代物理学中扮演着重要的角色。它们是黑洞热力学的天然领域，并出现在ADS/CFT通信中，该通信将这些几何与量子共形场理论联系起来，并体现了“全息术”，即一个区域内的物理由该区域边界上的其他物理编码。时空的几何受爱因斯坦的广义相对论支配。爱因斯坦的方程式包含了丰富多样的AH和AADS几何结构。有些没有物质，也没有黑洞，但却有质量-确实是负质量！-并且具有不寻常的形状或拓扑。这项研究计划试图研究这些迷人的几何图形。在其他问题中，它会问质量什么时候可以是负的，它可以负到什么程度，以及这与拓扑有什么关系？它还将询问在由其他几何方程控制的几何中是否存在类似的结构，即四阶偏微分方程。*几何可以变形，使其更平滑和更对称。这种变形是数学家和物理学家的重要工具。这个建议的一部分涉及到对被Ricci流变形的AH几何的研究，Ricci流是最近用来证明Poincaré猜想的一种方法。该提议旨在确定由Ricci流变形的AH几何的详细演化。*该提议的另一部分涉及爱因斯坦方程的推广，其中Ricci曲率张量被一个更一般的对象--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，还是也被其他更一般的几何所共享。*简而言之，该提议通过利用渐近双曲流形、几何流动和具有Bakry-émery-Ricci下界的密度流形的最新数学进展，寻求物理中重要问题的答案，无论是在ADS/CFT通信中还是在广义相对论中。

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.