On Chern's conjecture for minimal hypersurfaces in spheres

Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a compact minimal hypersurface in ${\mathbb{S}}^{n+1}$ whose squared length of the second fundamental form satisfies $0\le S-n\le \frac{n}{18}$, then $S\equiv n$ and $M$ is a Clifford torus.