On stable compact minimal submanifolds of Riemannian product manifolds

TL;DR

This paper classifies stable compact minimal submanifolds in certain Riemannian product manifolds, extending previous results and showing nonexistence under specific curvature conditions.

Contribution

It generalizes classification results for stable minimal submanifolds to broader curvature bounds in product manifolds involving hypersurfaces.

Findings

Classification theorem for stable compact minimal submanifolds under curvature bounds

Extension of previous results to new curvature conditions

Nonexistence of stable compact minimal submanifolds in certain hypersurfaces

Abstract

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K_{M_1}\leq 1.$ 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 ($m\geq3$) complete hypersurface $M$ in the Euclidean space, if the sectional curvature $K_{M}$ of $M$ satisfies $\frac{1}{\sqrt{m+1}}\leq K_{M}\leq 1$, then we conclude that there exist no stable compact minimal submanifolds in $M$.