Given a metric g on the 2-sphere S^2, we introduce a new invariant of g that is motivated by the quasi-local mass problem in general relativity. The new invariant is defined as the supremum of the total boundary mean curvature of all compact 3-Riemannian manifolds with nonnegative scalar curvature, with mean-convex boundary that is isometric to (S^2, g). The finiteness of this supremum follows from the earlier work of Wang-Yau and Shi-Tam on manifolds with a negative scalar curvature lower bound. We will discuss properties of this invariant and its relativistic implication. The talk is based on joint work with Christos Mantoulidis.

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