The biharmonic hypersurface flow and the Willmore flow in higher dimensions

TL;DR

This paper investigates the biharmonic hypersurface flow and Willmore flow in higher dimensions, establishing new inequalities and extending solutions, including solving an open problem for four-dimensional hypersurfaces and proving global existence.

Contribution

It introduces new Gagliardo-Nirenberg inequalities for hypersurfaces and applies them to extend solutions and prove global existence in higher dimensions, including a specific case for n=4.

Findings

Extended the maximal existence time for biharmonic flow in higher dimensions.

Solved an open problem for biharmonic hypersurface flow when n=4.

Proved global existence of 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 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.