A gap theorem of self-shrinkers

Abstract

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker with polynomial volume growth in Euclidean space ${\mathbb{R}}^{n+1}$ is isometric to either ${\mathbb{R}}^n$, $S^n(\sqrt{n})$, or ${\mathbb{R}}^{n-m}\times S^m(\sqrt{m})$, $1\le m\le n-1$, if the squared norm $S$ of the second fundamental form is constant and satisfies $S<(10/7)$.

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References in corpus (2)

• The rigidity theorems of self shrinkers
• Classification and rigidity of self-shrinkers in the Mean curvature flow