### 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) |
---|---|

Pages (from-to) | 271-284 |

Number of pages | 14 |

Journal | Communications in Mathematical Physics |

Volume | 292 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

**On a Localized Riemannian Penrose Inequality.** / Miao, Pengzi.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 292, no. 1, pp. 271-284. https://doi.org/10.1007/s00220-009-0834-0

}

TY - JOUR

T1 - On a Localized Riemannian Penrose Inequality

AU - Miao, Pengzi

PY - 2009/8

Y1 - 2009/8

N2 - 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 [9] and by Bray [5].

AB - 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 [9] and by Bray [5].

UR - http://www.scopus.com/inward/record.url?scp=70349488847&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349488847&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0834-0

DO - 10.1007/s00220-009-0834-0

M3 - Article

VL - 292

SP - 271

EP - 284

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -