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arXiv · 2404.04160

Optimal rigidity estimates for varifolds almost minimizing the Willmore energy

Yuchen Bi·Jie Zhou
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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 $\int_{\Sigma}|H|^2d\mu\le 16\pi(1+\delta^2)$ , 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 $\delta_0^2=2\pi$.

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Yuchen Bi, Jie Zhou. 2024-04-05. Optimal rigidity estimates for varifolds almost minimizing the Willmore energy. https://arxiv.org/abs/2404.04160

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