Rigidity of closed minimal hypersurfaces in S5

Abstract

Let $M^4\to S^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere $S^5$. In this paper, we prove that if the 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.