Area-minimizing regions with small volume in Riemannian manifolds with boundary

Mouhamed Moustapha Fall

Vol. 244 (2010), No. 2, 235–260

DOI: 10.2140/pjm.2010.244.235

Abstract

Given a domain Ω of a Riemannian manifold, we prove that regions minimizing the area (relative to Ω) are nearly the maxima of the mean curvature of ∂Ω when their volume tends to zero. We deduce some sharp local relative isoperimetric inequalities involving mean curvature comparisons.

Keywords

CMC surfaces, constant mean curvature, isoperimetric profile, isoperimetric region

Mathematical Subject Classification 2000

Primary: 53A10, 53C20, 53C21

Milestones

Received: 1 October 2008

Accepted: 13 February 2009

Published: 14 December 2009