New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces

Abstract

For any $n$-dimensional compact minimal hypersurface of the $(n+1)$-dimensional sphere, we have that the coordinate functions of the Gauss map are eigenfunctions of the stability operator associated with the eigenvalue $-n$. In this paper we consider minimal immersions from $\mbS^{k}\times\mbS^{\ell}\times\mbS^{1}$ to $\mbS^{k+\ell+2}$ of the form $ϕ(y,z,t)=\left(f(t) y, f_2(t) z, f_1(t)\right)$ and we explicitly show new eigenfunctions for the stability operator associated with the same eigenvalue. We also show that the stability index of these minimal immersions is at least $k\ell+3k+3\ell+8$.