On embedded minimal hypersurfaces in Sⁿ⁺¹ with symmetries

2020/10/30 by Changping Wang, Peng Wang, Wang, Changping +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.2010.16261

7 pages. Comments are welcome

arxiv created 2020/10/30 · openalex publication_date 2020/10/30 · arxiv updated 2020/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we generalize a characterization of the Clifford torus due to Ros. Let f:M→ Sⁿ⁺¹ be an embedded closed minimal hypersurface. Assume there are (n+2) great hyperspheres of Sⁿ⁺¹ perpendicular to each other, such that M is symmetric with respect to them. Let S denote the square of the length of the second fundamental form of f and let S=(1)/(Vol(M))∫_M Sd M be the average of S. Then S≥ n with equality holding if and only if f is the Clifford torus C_m,n-m. It can be rewritten as a Simons' type theorem: If 0≤ ∫_M (n-S)d M, then either S≡0 or S≡ n. This answers partially a conjecture by Perdomo. Moreover, the estimate of the Willmore energy of f is built: W(M)≥ n^(n)/(2)Vol(M).

Citations

• The scalar curvature of minimal hypersurfaces in spheres
• On Chernʼs problem for rigidity of minimal hypersurfaces in the spheres
• Moebius geometry of submanifolds in 𝕊 n
• Local Rigidity Theorems for Minimal Hypersurfaces
• Embedded minimal tori in S3 and the Lawson conjecture
• Minimal Varieties in Riemannian Manifolds
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Related

• A new characterization for Clifford hypersurfaces
• On Chern's conjecture for minimal hypersurfaces in spheres
• On the Uniqueness of Clifford Torus with Prescribed Isoperimetric Ratio
• Embedded minimal tori in S3 and the Lawson conjecture
• New result on Chern conjecture for minimal hypersurfaces and its application