A Rigidity Theorem for Asymptotically Flat Static Manifolds and its Applications

22.06.2023, 14:15-15:45  –  Zoom
Geometric Analysis, Differential Geometry and Relativity

Brian Harvie (National Taiwan University)

In general relativity, many physically and mathematically important questions concern the uniqueness of the Schwarzschild space. For example, the black hole uniqueness theorem states that the Schwarzschild space is the only asymptotically flat static manifold with stable minimal boundary, and similar uniqueness questions for photon surfaces and for static metric extensions have recently generated great interest.

In this talk, I will present a new approach to these questions that is based on a recently-discovered Minkowski inequality for asymptotically flat static manifolds. We prove that the equality is achieved only on coordinate spheres in the Schwarzschild space under natural boundary assumptions. From this, we derive several new uniqueness theorems for Schwarzschild. Notably, we establish global uniqueness of static metric extensions for the Bartnik data induced by Schwarzschild coordinate spheres in all dimensions less than 8. 

This is based on joint work Ye-Kai Wang of NYCU.

This talk is part of the seminar Geometric Analysis, Differential Geometry and Relativity organized by Carla Cederbaum (Uni Tübingen),  and Jan Metzger (Uni Potsdam) . To obtain the Zoom data please contact jan.metzger@uni-potsdam.de .