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

TL;DR

This paper investigates the supremum of total mean curvature on boundary surfaces of 3-manifolds with nonnegative scalar curvature, establishing properties and a variational analog of Brown-York mass relevant to general relativity.

Contribution

It introduces a new variational approach to Brown-York mass without requiring positive Gauss curvature on boundary spheres.

Findings

Finiteness of the supremum follows from Shi-Tam and Wang-Yau results.

Additivity property of the supremum is established.

Rigidity results for maximizers are proved.

Abstract

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 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 variational analog of Brown-York quasi-local mass without assuming that the boundary 2-sphere has positive Gauss curvature.