Surface diffusion flow of arbitrary codimension in space forms

Dong Pu and Hongwei Xu

Vol. 306 (2020), No. 1, 291–320

DOI: 10.2140/pjm.2020.306.291

Abstract

We first prove that a properly immersed static $n$-dimensional submanifold $\left(\Delta H=0\right)$ with restricted growth of the curvature at infinity in ${\mathbb{𝔽}}^{n+p}\left(c\right)\left(c\ge 0\right)$ is totally umbilical if the Willmore functional is pinched by a positive constant depending only on $n$. Secondly, we obtain a global rigidity theorem for Willmore surfaces in the sphere. Thirdly, we give a lower bound on the lifespan of the surface diffusion flow in ${\mathbb{𝔽}}^{2+p}\left(c\right)$. Finally, we get the longtime existence and convergence of the surface diffusion flow in ${\mathbb{𝔽}}^{2+p}\left(c\right)\left(c\ge 0\right)$ under the small initial Willmore energy condition.

Dedicated to Professor Duanzhuang Qian on his 110th anniversary.

Keywords

surface diffusion, Willmore surface, gap theorem, convergence theorem.

Mathematical Subject Classification 2010

Primary: 53C24, 53C44

Milestones

Received: 10 April 2019

Revised: 11 December 2019

Accepted: 20 December 2019

Published: 14 June 2020