A gap Theorem on closed self-shrinkers of mean curvature flow

Yuhang Zhao

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arXiv · 2503.00505

A gap Theorem on closed self-shrinkers of mean curvature flow

Yuhang Zhao

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Abstract

In this paper, we prove a pinching theorem for $n-$dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: $ | \vec{\uppercase\expandafter{\romannumeral2}} |^2 \le 1 +\frac{1}{10 \pi (n+2)}$, then it must be the standard sphere $S^{n}(\sqrt{n})$. This result may provide some evidence for the open problem 13.76 in \cite{andrews2022extrinsic}.

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Yuhang Zhao. 2025-03-01. A gap Theorem on closed self-shrinkers of mean curvature flow. https://arxiv.org/abs/2503.00505

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