Mathematical relativity and asymptotically hyperbolic manifolds

如果有两种常见的几何，那就是欧几里得的平面几何和球体的圆形几何。第三种是“鞍形”双曲几何，人们不太熟悉，但由于艺术家MC埃舍尔的作品而普及开来。这三个模型代表了恒定曲率几何，并表现出最大的对称性，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德西特（AAdS）时空，都是不对称的。它们在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舞台，出现在AdS/CFT对应中，它将这些几何形状与量子共形场论联系起来，并体现了“全息”，一种推测性的概念，即一个区域内的物理是由该区域边界上的其他物理编码的。

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.