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arXiv · 1212.6028

A gap theorem of self-shrinkers

Qing-Ming Cheng·Guoxin Wei
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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 $\mathbb{R}^{n-m}\times S^m (\sqrt{m})$, $1\leq m\leq n-1$, if the squared norm $S$ of the second fundamental form is constant and satisfies $S<(10/7)$.

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Qing-Ming Cheng, Guoxin Wei. 2012-12-25. A gap theorem of self-shrinkers. https://arxiv.org/abs/1212.6028

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