Geometry/topology Seminar
Thursday, March 9, 2017, 3:15pm, 128 Physics
Anna Sakovich (Uppsala University)
On the mass of asymptotically hyperbolic manifolds

Abstract:

A complete Riemannian manifold is called asymptotically hyperbolic if its ends are locally modeled on neighborhoods of infinity in hyperbolic space. This can be naturally interpreted through the notion of conformal compactification as introduced in general relativity in the early 1960s by Penrose. Since then, asymptotically hyperbolic manifolds have played an important role in conformal geometry, relativity, and string theory through the AdS/CFT correspondence. The notion of mass for asymptotically hyperbolic manifolds is more delicate than the concept of ADM mass in the asymptotically Euclidean setting, in the sense that it is not natural to isolate a `scalar' mass, but rather there is a family of mass-like invariants. For the Wang mass, defined by integrating a mass aspect density, there is a positive mass theorem on spin manifolds. In the general case, Andersson, Cai, and Galloway used an innovative technique based on the Witten-Yau BPS brane action to prove that this mass aspect cannot be everywhere negative. In this talk, we will discuss the proof of the fact that the mass itself is non-negative, which relies on the original ideas of Schoen and Yau and involves a blow-up analysis of the Jang equation. I will also report on joint work in progress with Mattias Dahl and Romain Gicquaud aimed at proving Penrose inequality in the asymptotically hyperbolic setting.

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