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


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.

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