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

Abstract

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere ${\mathbb{S}}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an equatorial 4-sphere, a product of spheres ${\mathbb{S}}^2(\frac{\sqrt{2}}{2})\times {\mathbb{S}}^2(\frac{\sqrt{2}}{2})$ or ${\mathbb{S}}^1(\frac{1}{2})\times {\mathbb{S}}^3(\frac{\sqrt{3}}{2})$, 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.

20 pages