Time：11:00-12:00, Thursday, June 5 2025
Venue：E4-233
Speaker: Min-Chun Hong, University of Queensland
Title：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 $n\geq 2$ 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 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. This is a joint work with Yu Fu and Gang Tian.
