A new pinching theorem for complete self-shrinkers and its generalization

Abstract: In this paper, we firstly verify that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}<sup>{n+1}$, and if the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|<sup>2-1\leq\frac{1}{18}$, then $|A|<sup>2\equiv1$ and $M$ is a round sphere or a cylinder. More generally, let $M$ be a complete $\lambda$-hypersurface with polynomial volume growth in $\mathbb{R}<sup>{n+1}$ with $\lambda\neq0$. Then we prove that there exists an positive constant $\gamma$, such that if $|\lambda|\leq\gamma$ and the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|<sup>2-\beta_\lambda\leq\frac{1}{18}$, then $|A|^2\equiv \beta_\lambda$, $\lambda&gt;0$ and $M$ is a cylinder. Here $$\beta_\lambda=\frac{1}{2}(2+\lambda<sup>2+|\lambda|\sqrt{\lambda<sup>2+4})$$.