On stable compact minimal submanifolds of Riemannian product manifolds

In this paper, we prove a classification theorem for stable compact minimal submanifolds of a Riemannian product of an m 1 -dimensional ( m 1 ≥ 3 ) hypersurface M 1 in Euclidean space and any Riemannian manifold M 2 , when the sectional curvature K M 1 of M 1 satisfies 1 m 1 − 1 ≤ K M 1 ≤ 1 . In particular, when the ambient space is an m -dimensional ( m ≥ 3 ) compact hypersurface M in 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 .