## Journal of Differential Geometry

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

### 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*| ≤ 16π*m*^{2}. 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.