On the stability of minimal cones in warped products

Abstract

In a seminal paper published in 1968, J. Simons proved that, for n ≤ 5, the Euclidean (minimal) cone CM, built on a closed, oriented, minimal and non totally geodesic hypersurface M n of $\mathbb{S}^{n + 1} $ is unstable. In this paper, we extend Simons’ analysis to warped (minimal) cones built over a closed, oriented, minimal hypersurface of a leaf of suitable warped product spaces. Then, we apply our general results to the particular case of the warped product model of the Euclidean sphere, and establish the unstability of CM, whenever 2 ≤ n ≤ 14 and M n is a closed, oriented, minimal and non totally geodesic hypersurface of $\mathbb{S}^{n + 1} $ .