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

摘要

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.