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

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.

Keywords

Parabolic equation / mean curvature flow / maximum principle (for tensor)

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