The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace

Let $M^n\subset {\mathbb{S}}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=\mathrm{\mid }h{\mathrm{\mid }}^2$ and constant $f_3=\mathrm{tr}{h}^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\ge 2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions $n=3,6,12,24$.