Brian Harvie (Columbia): Static spaces, geometric inequalities, and the uniqueness of the Schwarzschild metric

Static spaces are Riemannian manifolds which arise as constant time slices of static vacuum spacetimes in general relativity. The most familiar example of a static space is Schwarzschild space, which is the base of the Schwarzschild spacetime one dimension higher. The Schwarzschild spacetime contains both a black hole and a photon surface, and it is physically natural to ask if Schwarzschild is the only (asymptotically flat) static spacetime containing these astrophysical objects. These uniqueness questions may be posed as boundary value problems for static spaces. 
In this talk, I will present a new approach to these boundary value problems based on geometric inequalities. Specifically, by establishing rigidity within a Minkowski inequality for hypersurfaces in asymptotically flat static spaces, we may generalize black hole and photon surface uniqueness under weaker asymptotic assumptions and to higher dimensions. This approach is also connected to the study of quasi-spherical metrics, which arise elsewhere in mathematical general relativity.