Optimal rigidity estimates for varifolds almost minimizing the Willmore energy

Abstract

For an integral $2$-varifold $V=\underline{v}(\Sigma ,{\theta }_{\ge 1})$ in ${\mathbb{R}}^n$ with generalized mean curvature $H\in L^2$ such that $\mu ({\mathbb{R}}^n)=4\pi$ and ${\int }_{\Sigma }|H{|}^{2}d\mu \le 16\pi (1+{\delta }^2)$ , we show that $\Sigma$ is $W^{2,2}$ close to the standard embedding of the round sphere in a quantitative way when $\delta <{\delta }_0\ll 1$. For $n=3$, we prove that the sharp constant is ${\delta }_0^2=2\pi$.

24 pages