The second gap on complete self-shrinkers

Abstract

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker in Euclidean space ${\mathbb{R}}^{n+1}$ is isometric to either ${\mathbb{R}}^n$, $S^n(\sqrt{n})$, or $S^k(\sqrt{k})\times {\mathbb{R}}^{n-k}$, $1\le k\le n-1$, if the squared norm $S$ of the second fundamental form, $f_3$ are constant and $S$ satisfies $S<1.83379$. We should remark that the condition of polynomial volume growth is not assumed.