On stable compact minimal submanifolds of Riemannian product manifolds

Abstract

In this paper, we prove a classification theorem for stable compact minimal submanifolds of a Riemannian product of an m1-dimensional (m1≥3) hypersurface M1 in Euclidean space and any Riemannian manifold M2, when the sectional curvature KM1 of M1 satisfies 1m1−1≤KM1≤1. In particular, when the ambient space is an m-dimensional (m≥3) compact hypersurface M in Euclidean space, if the sectional curvature KM of M satisfies 1m+1≤KM≤1, then we conclude that there exist no stable compact minimal submanifolds in M.