On the capacity of surfaces in manifolds with nonnegative scalar curvature

Hubert Bray, Pengzi Miao

Research output: Contribution to journalArticle

18 Scopus citations

Abstract

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 ∞. Even in the special case of ℝ3, 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.

Original languageEnglish (US)
Pages (from-to)459-475
Number of pages17
JournalInventiones Mathematicae
Volume172
Issue number3
DOIs
StatePublished - Jun 1 2008
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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