Rigidity of submanifolds with parallel mean curvature in space froms

2011/05/15 by Hongwei Xu, Xu, Hong-Wei, Juan-Ru Gu +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1105.2920

openalex publication_date 2011/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be an n(≥3)-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form F^n+p(c) with c+H²>0, where H is the mean curvature of M. We prove that if the Ricci curvature of M satisfies Ric_M≥(n-2)(c+H²), then M is either a totally umbilic sphere, the Clifford hypersurface S^m((1)/(√(2(c+H²))))× S^m((1)/(√(2(c+H²)))) in Sⁿ⁺¹((1)/(√(c+H²))) with n=2m, or ℂP²(4/3(c+H²)) in S⁷((1)/(√(c+H²))). In particular, if Ric_M>(n-2)(c+H²), then M is a totally umbilic sphere.

Citations

• On Yau rigidity theorem for minimal submanifolds in spheres
• Local Rigidity Theorems for Minimal Hypersurfaces
• Minimal Varieties in Riemannian Manifolds

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