Closed minimal hypersurfaces in ${\mathbb{S}}^5$ with constant $S$ and $A_3$

In this paper, we prove that a closed minimally immersed hypersurface $M^4\subset {\mathbb{S}}^5$ with constant $S:={\sum }_{i=1}^4{\lambda }_i^2$ and $A_3:={\sum }_{i=1}^4{\lambda }_i^3$ whose scalar curvature $R_M$ is nonnegative must be isoparametric. Moreover, $S$ can only be $0,4,$ and $12.$ That is $M^4$ is either an equatorial $4$-sphere, a clifford torus, or a Cartan’s minimal hypersurface.