New result on Chern conjecture for minimal hypersurfaces and its application

We verify that if $M$ is a compact minimal hypersurface in ${\mathbb{S}}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$,then$|A|^2\equiv n$and$M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in ${\mathbb{R}}^{n+1}$ whose equation is given by (\ref{selfshr}), and if the squared length of the second fundamental form of $M$ satisfies $0\le \mathrm{\mid }A{\mathrm{\mid }}^2-1\le \frac{1}{21}$, then $\mathrm{\mid }A{\mathrm{\mid }}^2\equiv 1$ and $M$ is a round sphere or a cylinder. Our results improve the rigidity theorems due to Q. Ding and Y. L. Xin \cite{DX1,DX2}.