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Asymptotically flat static spaces are Riemannian manifolds that correspond to static vacuum spacetimes in general relativity. The most important example is the Schwarzschild space, a rotationally symmetric Riemannian manifold corresponding to the Schwarzschild spacetime. A number of important questions about the uniqueness of the Schwarzschild spacetime may be posed as rigidity questions for AF static spaces. These include the famous static black hole uniqueness theorems of Israel and Bunting/Masood-ul-Alam as well as the more recent uniqueness theorems for static spacetimes containing photon surfaces.
In this talk, I will present a new approach to these questions that is based on a Minkowski-type inequality for AF static spaces. Like the Minkowski inequality for convex hypersurfaces in Euclidean space, this inequality gives a bound from below on the total mean curvature of the boundary of the manifold. First, I will characterize rigidity within this inequality, showing under suitable boundary assumptions that the equality is achieved only by rotationally symmetric regions of Schwarzschild space. As an application, I will show uniqueness of suitably-defined static metric extensions for the Bartnik data of Schwarzschild coordinate spheres. This talk is based on joint work with Ye-Kai Wang of NYCU.