The author studies the L2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λi ≥ 0 and co, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.
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.
