Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

2025/12/09 by Ge, Jianquan, Tan, Huixin, Yan, Wenjiao +1
Mathematics · #53C20 #53C24 #53C40 #Advanced Harmonic Analysis Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2512.08342

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that for an n-dimensional closed minimal Willmore hypersurface Mⁿ with constant scalar curvature in the unit sphere \mathbbSⁿ⁺¹, the squared norm S of the second fundamental form of Mⁿ satisfies S\geqslant n+\frac4n+9-√4 n²+60 n+812 if S>n. This proves, in the approximate sense, the Chern conjecture about the second gap (S\geqslant 2n if S>n), which will be fully verified under a further inequality condition about the 4-th mean curvature.

Citations

• On the Chern conjecture for isoparametric hypersurfaces
• A sufficient condition for a hypersurface to be isoparametric
• Chern conjecture and isoparametric hypersurfaces
• Chern conjecture on minimal hypersurfaces
• Minimal Varieties in Riemannian Manifolds
• Local Rigidity Theorems for Minimal Hypersurfaces

Related

• Chern conjecture on minimal hypersurfaces
• New result on Chern conjecture for minimal hypersurfaces and its application
• On the Chern conjecture for isoparametric hypersurfaces
• Some results on Chern's problem
• On Chern's conjecture for minimal hypersurfaces in spheres