An improved result on Chern conjecture

Let $X:M^n\to {\mathbb{S}}^{n+1}(1)$ be a closed minimal hypersurface with constant scalar curvature. Chern conjecture says that a closed minimal hypersurface with constant scalar curvature, $S\ge 2n$ if $S>n$, where $S$ is the squared norm of the second fundamental form. As a partial result, Peng-Terng \cite{pt1, pt2}, Yang-Cheng \cite{yc1, yc2, yc3} and Suh-Yang \cite{sy} obtained that if $S>n$, then $S>n+{\displaystyle \frac{3}{n}}$. Recently, Chen \cite{c} has proved $S>n+\frac{26}{59}n$ if $S>n$. Since resolving Chern conjecture is a hard problem, as a midway for resolving Chern conjecture, one wants to prove $S>n+\frac{1}{2}n$ if $S>n$. In this paper, for $4\le n\le 13$, we give a positive solution for this midway problem. For general $n$, we prove that for a complete minimal hypersurface with constant scalar curvature, $$S>\frac{10000000}{6869671}n>1.4556737869n>n+{\displaystyle \frac{221}{485}}n$$ if $S>n$, which is better than the result of Chen \cite{c}.