On stable compact minimal submanifolds

Abstract

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.2010 Mathematics Subject Classification. Primary 53C40, 53C42.

On stable compact minimal submanifolds of Riemannian product manifolds

“…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 ≥ 3) complete hypersurface M in the Euclidean space, if the sectional curvature KM of M satisfies 1 √ m+1 ≤ KM ≤ 1, then we conclude that there exist no stable compact minimal submanifolds in M .…”

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

“…We use the same idea as in [3], [5], [8] and [9], etc., that is, taking the normal components of parallel vector fields of the Euclidean space where M 1 sits as the test sections.…”

“…They proved that Theorem 1.2 (Torralbo and Urbano, [9]). Let M be any Riemannian manifold and Φ = (φ, ψ) : Σ → S m (r) × M be a minimal immersion of a compact n-manifold Σ, n ≥ 2, satisfying either m ≥ 3 or m = 2 and Φ is a hypersurface.…”

“…It is remarkable that very recently Torralbo and Urbano (see [9]) proved a classification theorem for the stable compact minimal submanifolds of the Riemannian product of a sphere S m (r) and any Riemannian manifold M . They proved that Theorem 1.2 (Torralbo and Urbano, [9]).…”