The second gap for self-shrinkers with constant norm of the second fundamental form

Abstract

Let $X:M^n\to {\mathbb{R}}^{n+1}$ be a complete self-shrinker with constant squared norm of the second fundamental form $S$. In this paper, we prove that if $S\le \frac{10}{7}$, then $S=1$ or $S=0$, and the self-shrinker is isometric to either a round sphere ${\mathbb{S}}^n(\sqrt{n})$ with the center at the origin, or a cylinder ${\mathbb{S}}^k(\sqrt{k})\times {\mathbb{R}}^{n-k},1\le k\le n-1$, or a plane through the origin.

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