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

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