Let Ω be a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary ∂Ω is the disjoint union of two pieces: ΣH and ΣO, where ΣH consists of the unique closed minimal surfaces in Ω and ΣO is metrically a round sphere. We obtain an inequality relating the area of ΣH to the area and the total mean curvature of ΣO. Such an Ω may be thought of as a region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. The inequality we establish has close ties with the Riemannian Penrose Inequality, proved by Huisken and Ilmanen  and by Bray .
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics