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 talk, I will present recent joint work with M. Eichmair that resolves this conjecture: The exterior mass of an asymptotically flat support surface S with nonnegative mean curvature is bounded in terms of the area of the outermost free boundary minimal surface supported on S. If equality holds, then the exterior surface of S is a half-catenoid. In particular, we obtain 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.

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