The Penrose inequality in extrinsic geometry

TL;DR

This paper proves an extrinsic geometric analogue of the Penrose inequality, relating the exterior mass of a surface to the area of a minimal boundary, with implications for characterizing catenoids among minimal surfaces.

Contribution

It resolves G. Huisken's conjecture by establishing a new extrinsic Penrose inequality and characterizes the catenoid via this inequality in the context of minimal surfaces.

Findings

Exterior mass is bounded below by the square root of boundary area divided by pi.

Equality case characterizes the unbounded component as a half-catenoid.

Introduces a nondecreasing quantity related to minimal capillary surfaces as contact angle varies.

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.