The Penrose inequality in extrinsic geometry

Abstract

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this paper, we resolve this conjecture and show that the exterior mass $m$ of an asymptotically flat support surface $S\subset\mathbb{R}^3$ with nonnegative mean curvature and outermost free boundary minimal surface $D$ is bounded in terms of $$ m\geq \sqrt{\frac{|D|}{\pi}}. $$ If equality holds, then the unbounded component of $S\setminus \partial D$ is a half-catenoid. In particular, this extrinsic Penrose inequality leads to a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on $S$ that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases.