The biharmonic hypersurface flow and the Willmore flow in higher dimensions

Abstract

The biharmonic flow of hypersurfaces $M^n$ immersed in the Euclidean space ${\mathbb{R}}^{n+1}$ for $n\ge 2$ 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 posed by Bernard et al. (2019) 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.