The volume-preserving mean curvature flow in Euclidean space

Haozhao Li

Vol. 243 (2009), No. 2, 331–355

DOI: 10.2140/pjm.2009.243.331

Abstract

We study the convergence of the volume-preserving mean curvature flow of hypersurfaces in Euclidean space under some initial integral pinching conditions. We prove that if the traceless second fundamental form is sufficiently small, the flow will exist for all time and converge exponentially fast to a round sphere.

Keywords

volume-preserving mean curvature flow, stability

Mathematical Subject Classification 2000

Primary: 53C44, 53A10

Milestones

Received: 1 July 2008

Revised: 25 May 2009

Accepted: 31 July 2009

Published: 1 December 2009