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.