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arXiv · 2505.19727

The biharmonic hypersurface flow and the Willmore flow in higher dimensions

Yu Fu·Min-Chun Hong·Gang Tian
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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 in \cite{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.

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Yu Fu, Min-Chun Hong, Gang Tian. 2025-05-26. The biharmonic hypersurface flow and the Willmore flow in higher dimensions. https://arxiv.org/abs/2505.19727

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