A note on the Chern Conjecture in dimension four

Abstract: Let $M<sup>4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in $\mathbb{S}<sup>{5}$. If $H_3^2\leq \frac{1}{2}.$ and Gauss-Kronecker curvature $K_M$ satisfies $K_M\leq1$ $($or $K_M\leq\frac{S^2}{144}$)$, then$M^4$ is isoparametric.