#### Vol. 306, No. 1, 2020

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Surface diffusion flow of arbitrary codimension in space forms

### Dong Pu and Hongwei Xu

Vol. 306 (2020), No. 1, 291–320
##### 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 ${\mathbb{𝔽}}^{n+p}\left(c\right)\phantom{\rule{0.28em}{0ex}}\left(c\ge 0\right)$ 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 ${\mathbb{𝔽}}^{2+p}\left(c\right)\phantom{\rule{0.28em}{0ex}}\left(c\ge 0\right)$ 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