Rigidity of closed minimal hypersurfaces in S5

Pengpeng Cheng; Tongzhu Li

doi:10.1016/j.difgeo.2025.102252

Rigidity of closed minimal hypersurfaces in S⁵

Pengpeng Cheng, Tongzhu Li^*

^*Corresponding author for this work

School of Mathematics and Statistics

Beijing Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

Let M⁴→S⁵ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere S⁵. In this paper, we prove that if the 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.

Original language	English
Article number	102252
Journal	Differential Geometry and its Application
Volume	100
DOIs	https://doi.org/10.1016/j.difgeo.2025.102252
Publication status	Published - Sept 2025

Keywords

Chern conjecture
Isoparametric hypersurfaces
Minimal hypersurfaces
The second fundamental form

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10.1016/j.difgeo.2025.102252

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abstract = "Let M4→S5 be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere S5. In this paper, we prove that if the 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.",

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AB - Let M4→S5 be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S in a 5-dimensional sphere S5. In this paper, we prove that if the 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.

KW - Chern conjecture

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KW - Minimal hypersurfaces

KW - The second fundamental form

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Rigidity of closed minimal hypersurfaces in S⁵

Abstract

Keywords

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