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

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