Frontiers of Mathematics in China >

2012 , Vol. 7 >Issue 4: 717 - 723

DOI: https://doi.org/10.1007/s11464-012-0218-1

RESEARCH ARTICLE

Evolution of hypersurfaces by powers of mean curvature minus an external force field

  • Yannan LIU
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  • Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China

Received date: 16 Sep 2010

Accepted date: 04 May 2012

Published date: 01 Aug 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

In this paper, we study the evolution of hypersurfaces by powers of mean curvature minus an external force field. We prove that when the power is 2, the flow has a long-time smooth solution for all time under some conditions. Those conditions are that the second fundamental form on the initial submanifolds is not too large, the external force field, with its any order derivatives, is bounded, and the field is convex with its eigenvalues satisfying a pinch inequality.

Key words： Parabolic equation; mean curvature flow; maximum principle (for tensor)

Cite this article

Yannan LIU . Evolution of hypersurfaces by powers of mean curvature minus an external force field[J]. Frontiers of Mathematics in China, 2012 , 7(4) : 717 -723 . DOI: 10.1007/s11464-012-0218-1

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