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

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}.