2024/11/04 by Michael Eichmair, Eichmair, Michael, Thomas Koerber +1
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2411.02113
openalex publication_date 2024/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊂ℝ3 with nonnegative mean curvature and outermost free boundary minimal surface D is bounded in terms of m≥ √\frac|D|π. If equality holds, then the unbounded component of S∖ ∂ 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.