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 $S^k (\sqrt {k})\times \mathbb {R}^{n-k}$, $1\leq k\leq n-1$, 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.