Self-Shrinkers With Second Fundamental Form of Constant Length

Abstract

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in ${\mathbb{R}}^3$ with second fundamental form of constant length must be a generalized cylinder ${\mathbb{S}}^k\times {\mathbb{R}}^{2-k}$ for some $k\le 2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

6 pages

References in corpus (1)

• Self-shrinkers with a rotational symmetry

Cited by in corpus (1)

• Gap and rigidity theorems of $\lambda$-hypersurfaces