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-M1 of M-1 satisfies 1/root m(1)-1 <= K-M1 <= 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/root m+1 <= K-M <= 1, then we conclude that there exist no stable compact minimal submanifolds in M. (c) 2013 Elsevier Inc. All rights reserved.