Complete self-shrinkers with bounded the second fundamental form in ${\mathbb{R}}^{n+1}$

Abstract

Let $X:M^n\to {\mathbb{R}}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\le S<C$ for some constant $C$, then $S=1$. Further we classify the $n$-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in ${\mathbb{R}}^{n+1}$, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.

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