Stability of Homogeneous minimal hypersurfaces in the Page space and $Y^{p,q}$ Sasaki-Einstein manifolds

We investigate the stability of homogeneous minimal submanifolds in two families of closed Einstein manifolds, the Page space ${\mathbb{C}\mathbb{P}}^2\mathrm{\#}\overline{{\mathbb{C}\mathbb{P}}^2}$ and the Sasaki-Einstein spaces $Y^{p,q}$, which are equipped with cohomogeneity-one Einstein metrics admitting the isometric action of $SU(2)\times U(1)$ and $U(1)\times U(1)\times SU(2)$ respectively. We determine all the homogeneous, minimal hypersurfaces and explicitly compute the spectrum of their associated stability operators and determine their index.