On a Localized Riemannian Penrose Inequality

Research output: Contribution to journalArticle

8 Citations (Scopus)

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: 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].

Original languageEnglish (US)
Pages (from-to)271-284
Number of pages14
JournalCommunications in Mathematical Physics
Volume292
Issue number1
DOIs
StatePublished - Aug 2009
Externally publishedYes

Fingerprint

curvature
minimal surfaces
unions
Nonnegative Curvature
Manifolds with Boundary
Total curvature
Scalar Curvature
Minimal surface
Tie
General Relativity
Mean Curvature
Slice
Black Holes
horizon
Riemannian Manifold
relativity
Horizon
Disjoint
Union
Space-time

ASJC Scopus subject areas

  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

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

In: Communications in Mathematical Physics, Vol. 292, No. 1, 08.2009, p. 271-284.

Research output: Contribution to journalArticle

@article{2f8bc8fe9da048feb3eda15e2a9f30e3,
title = "On a Localized Riemannian Penrose Inequality",
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: 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].",
author = "Pengzi Miao",
year = "2009",
month = "8",
doi = "10.1007/s00220-009-0834-0",
language = "English (US)",
volume = "292",
pages = "271--284",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "1",

}

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 -