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.

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