The biharmonic hypersurface flow and the Willmore flow in higher dimensions

TL;DR

This work develops a higher-dimensional framework for fourth-order curvature flows by proving new hypersurface Gagliardo-Nirenberg inequalities via the Michael-Simon-Sobolev inequality. Using local energy estimates and a covering argument, the authors extend short-time existence for the biharmonic flow to longer times and establish global existence in higher dimensions under a ballwise smallness condition on the second fundamental form $|A|$, solving the $n=4$ case of a prior problem. They further translate these techniques to the Willmore flow, demonstrating global existence and long-time convergence to a Willmore configuration in higher dimensions. Overall, the paper provides a cohesive energy-method toolkit for controlling curvature via local energy criteria in high-dimensional, fourth-order geometric flows.

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.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Theorem 1.3
• Remark 2
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof