  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 $k\leq2$. Moreover, we prove a
gap theorem for smooth self-shrinkers in all dimensions.
