Optimal rigidity estimates for varifolds almost minimizing the Willmore energy

Abstract:

For an integral $2$-varifold $V=\underline {v}(\Sigma ,\theta _{\geqslant 1})$ in $\mathbb {R}^n$ with generalized mean curvature $H\in L^2$ such that $\mu (\mathbb {R}^n)=4\pi$ and $\int _{\Sigma }|H|^2d\mu \leqslant 16\pi (1+\delta ^2)$, 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 $\delta _0^2=2\pi$.