Long-time existence of mean curvature flow with external force fields

Huai-Yu Jian and Yannan Liu

Vol. 234 (2008), No. 2, 311–324

DOI: 10.2140/pjm.2008.234.311

Abstract

We study the evolution of submanifolds moving by mean curvature and an external force field. We prove flow has a long-time smooth solution for all time under almost optimal conditions. Those conditions are that the second fundamental form on the initial submanifolds is not too large, the external force field and all of it derivatives are bounded, and the field is convex with its eigenvalues satisfying a pinch inequality.

Keywords

parabolic equation, mean curvature flow, maximum principle for tensors

Mathematical Subject Classification 2000

Primary: 58J35, 53A07, 35K45

Secondary: 53A10, 35K55, 58J15

Milestones

Received: 26 April 2007

Revised: 10 September 2007

Accepted: 29 October 2007

Published: 1 February 2008