On stable compact minimal submanifolds of Riemannian product manifolds

2012/09/28 by Hang Chen, Xianfeng Wang, Chen, Hang +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.1209.6400

11 pages

arxiv created 2012/09/28 · openalex publication_date 2012/09/28 · arxiv updated 2012/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an m₁-dimensional (m₁≥3) hypersurface M₁ in the Euclidean space and any Riemannian manifold M₂, when the sectional curvature K_M1 of M₁ satisfies (1)/(√(m₁-1))≤ K_M1≤ 1. This gives a generalization to the results of F. Torralbo and F. Urbano [9], where they obtained a classification theorem for the stable minimal submanifolds of the Riemannian product of a sphere and any Riemannian manifold. In particular, when the ambient space is an m-dimensional (m≥3) complete hypersurface M in the Euclidean space, if the sectional curvature K_M of M satisfies (1)/(√(m+1))≤ K_M≤ 1, then we conclude that there exist no stable compact minimal submanifolds in M.

Citations

• Minimal Varieties in Riemannian Manifolds
• On stable compact minimal submanifolds

Related

• On stable compact minimal submanifolds
• On minimal hypersurfaces in Euclidean spaces and Riemannian manifolds
• The geometry of stable minimal surfaces in metric Lie groups
• Stable submanifolds in the product of projective spaces
• A conformal invariant and its application to the nonexistence of minimal submanifolds