On a localized Riemannian Penrose inequality

Pengzi Miao

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21 Citations (Scopus)


Let Omega be a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary partial derivative Omega is the disjoint union of two pieces: Sigma(H) and Sigma(O), where Sigma(H) consists of the unique closed minimal surfaces in Omega and Sigma(O) is metrically a round sphere. We obtain an inequality relating the area of Sigma(H) to the area and the total mean curvature of Sigma(O). Such an Omega 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 [9] and by Bray [5].
Original languageEnglish
Pages (from-to)271 - 284
Number of pages14
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - 2009

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