### Abstract

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: Σ_{H} and Σ_{O}, where Σ_{H} consists of the unique closed minimal surfaces in Ω and Σ_{O} is metrically a round sphere. We obtain an inequality relating the area of Σ_{H} to the area and the total mean curvature of Σ_{O}. 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 [9] and by Bray [5].

Original language | English (US) |
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Pages (from-to) | 271-284 |

Number of pages | 14 |

Journal | Communications in Mathematical Physics |

Volume | 292 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics