Journal of Differential Geometry

The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality

Gerhard Huisken and Tom Ilmanen

Abstract

Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N| ≤ 16πm2. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch's monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik's gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.

Article information

Source
J. Differential Geom. Volume 59, Number 3 (2001), 353-437.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090349447

Digital Object Identifier
doi:10.4310/jdg/1090349447

Mathematical Reviews number (MathSciNet)
MR1916951

Zentralblatt MATH identifier
1055.53052

Citation

Huisken, Gerhard; Ilmanen, Tom. The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality. J. Differential Geom. 59 (2001), no. 3, 353--437. doi:10.4310/jdg/1090349447. https://projecteuclid.org/euclid.jdg/1090349447.


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