Quasi-spherical metrics and the static Minkowski inequality

Abstract: We prove that equality in the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space. To show this, we establish uniqueness of quasi-spherical static metrics: rotationally symmetric regions of Schwarzschild are the only static vacuum metrics which are quasi-spherical with vanishing shear vector. In addition, we observe that the static Minkowski inequality extends to all dimensions for a connected boundary and to asymptotically flat static manifolds of any positive decay order. Altogether, this yields a robust rigidity criterion for Schwarzschild space. Using this criterion, we recover Israel's static black hole uniqueness theorem under this mild decay assumption. Likewise, the uniqueness theorems for photon surfaces and static metric extensions from the prequel extend to all dimensions under these weaker asymptotics. Finally, as a notable by-product of our analysis, we establish regularity of weak inverse mean curvature flow in asymptotically flat manifolds -- that is, a weak IMCF is eventually smooth in an arbitrary asymptotically flat background.