On the stability of minimal cones in warped products

In a seminal paper published in $1968$, J. Simons proved that, for $n\le 5$, 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 {\em 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 $2\le n\le 14$ and $M^n$ is a closed, oriented, minimal and non totally geodesic hypersurface of ${\mathbb{S}}^{n+1}$.