Classification of closed minimal hypersurfaces with constant scalar curvature in $\mathbb{S}^5$

Abstract: In this paper, we prove that any closed minimal hypersurface $M<sup>4$ in the $5$-dimensional unit sphere $\mathbb{S}<sup>5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M<sup>4$ is either an equatorial 4-sphere, a product of spheres $$\mathbb{S} ^{{2}(\frac{\sqrt{2}}{2})} \times \mathbb{S} <sup>{2}(\frac{\sqrt{2}}{2})$$ or $$\mathbb{S} ^{{1}(\frac{1}{2})} \times \mathbb{S} <sup>{3}(\frac{\sqrt{3}}{2})$$, or a Cartan's minimal hypersurface. In particular, the value of the squared norm of the second fundamental form $S$ can only be 0, 4, or 12. This result strongly supports Chern's conjecture.