The biharmonic flow of hypersurfaces Mⁿ immersed in the Euclidean space R^n+1 for n 2 is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon-Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem in BWW on the biharmonic hypersurface flow for n=4. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions.
Fu et al. (Mon,) studied this question.