On stable compact minimal submanifolds of Riemannian product manifolds

Abstract

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\ge 3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\le K_{M_1}\le 1.$ 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 ($m\ge 3$) complete hypersurface $M$ in the Euclidean space, if the sectional curvature $K_M$ of $M$ satisfies $\frac{1}{\sqrt{m+1}}\le K_M\le 1$, then we conclude that there exist no stable compact minimal submanifolds in $M$.

11 pages

References in corpus (1)

• On stable compact minimal submanifolds