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Total mean curvature, scalar curvature, and a variational analog of Brown-York mass

Banff International Research Station for Mathematical Innovation and Discovery

We discuss the supremum of the total boundary mean curvature of compact, mean- convex 3-manifolds with nonnegative scalar curvature, with prescribed intrinsic boundary metric. We establish an additivity property for the supremum and exhibit rigidity for maximizers as- suming the supremum is attained. When the boundary consists of topological 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a varia- tional analog of the Brown-York mass without assuming the boundary surface has positive Gauss curvature. This is a joint work with Christos Mantoulidis.

Mathematics; Relativity and gravitational theory; Differential geometry

video/mp4

Author affiliation: University of Miami

2017-02-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60345

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/