On stable compact minimal submanifolds of Riemannian product manifolds

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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 Kₘ₁ of M₁ satisfies 1√m₁-1≤ Kₘ₁≤ 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 Kₘ of M satisfies 1√m+1≤ Kₘ≤ 1, then we conclude that there exist no stable compact minimal submanifolds in M.