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.

Keywords

Mathematical Subject Classification 2010

Milestones

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