2007/07/23 by Hubert Bray, Pengzi Miao · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #gr-qc #math.DG #msc:53C20 #msc:83C99
paper · pdf · doi:10.1007/s00222-007-0102-x
published as Invent. math. 172, 459-475 (2008) · 18 pages
arxiv created 2007/07/23 · openalex publication_date 2008/01/08 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.