Convergence of axially symmetric volume-preserving mean curvature flow

Maria Athanassenas and Sevvandi Kandanaarachchi

Vol. 259 (2012), No. 1, 41–54

DOI: 10.2140/pjm.2012.259.41

Abstract

We study the convergence of axially symmetric hypersurfaces evolving by volume-preserving mean curvature flow. Assuming the surfaces do not develop singularities along the axis of rotation at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that the flow converges to a hemisphere, when the initial hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.

Keywords

volume-preserving mean curvature flow, mean curvature flow with constraints

Mathematical Subject Classification 2010

Primary: 53C44

Secondary: 35K93

Milestones

Received: 13 September 2011

Revised: 20 March 2012

Accepted: 23 March 2012

Published: 31 August 2012