Rigidity of closed minimal hypersurface in $\mathbb{S}^5$

Abstract

Let $M^4\to \mathbb{S}^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ in a $5$-dimensional sphere $\mathbb{S}^5$. In this paper, we prove that if $3$-mean curvature $H_3$ and the number $g$ of the distinct principal curvatures are constant, then $M^4$ is an isoparametric hypersurface, and the value of $S$ can only be $0, 4, 12$. This result supports Chern Conjecture.