Rigidity of closed minimal hypersurface in \mathbbS⁵

2024/10/25 by Pengpeng Cheng, Cheng, Pengpeng, Tongzhu Li +1
Mathematics · #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2410.19531

Abstract

Let M⁴→ \mathbbS⁵ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere \mathbbS⁵. In this paper, we prove that if 3-mean curvature H₃ and the number g of the distinct principal curvatures are constant, then M⁴ is an isoparametric hypersurface, and the value of S can only be 0, 4, 12. This result supports Chern Conjecture.

Related

• An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature
• Closed minimal hypersurfaces in \mathbb S⁵ with constant S and A₃
• Minimal Hypersurfaces with constant scalar curvature in S⁶
• Strong Rigidity of Closed Minimal Hypersurfaces in Euclidean Spheres
• Chern conjecture on minimal hypersurfaces