The inverse mean curvature flow and the Riemannian Penrose Inequality

Huisken, Gerhard; Ilmanen, Tom

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The inverse mean curvature flow and the Riemannian Penrose Inequality

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Huisken, Gerhard
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Huisken, G., & Ilmanen, T. (2001). The inverse mean curvature flow and the Riemannian Penrose Inequality. Journal of Differential Geometry, 59(3), 353-437.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-5581-4

Abstract

LetM 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.