Einstein Hypersurfaces of Warped Product Spaces

We consider Einstein hypersurfaces of warped products $I\times_\omega\mathbb Q_\epsilon^n,$ where $I\subset\mathbb R$ is an open interval and $\mathbb Q_\epsilon^n$ is the simply connected space form of dimension $n\ge 2$ and constant sectional curvature $\epsilon\in\{-1,0,1\}.$ We show that, for all $c\in\mathbb R$ (resp. $c>0$), there exist rotational hypersurfaces of constant sectional curvature $c$ in $I\times_\omega\mathbb H^n$ and $I\times_\omega\mathbb R^n$ (resp. $I\times_\omega\mathbb S^n$), provided that $\omega$ is nonconstant. We also show that the gradient $T$ of the height function of any Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$ (if nonzero) is one of its principal directions. Then, we consider a particular type of Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$ with non vanishing $T$ -- which we call ideal -- and prove that such a hypersurface $\Sigma$ has either precisely two or precisely three distinct principal curvatures everywhere. We show that, in the latter case, there exist such a $\Sigma$ for certain warping functions $\omega,$ whereas in the former case, $\Sigma$ is necessarily of constant sectional curvature and rotational, regardless the warping function $\omega.$ We also characterize ideal Einstein hypersurfaces of $I\times_\omega\mathbb Q_\epsilon^n$ with no vanishing angle function as local graphs on families of isoparametric hypersurfaces of $\mathbb Q_\epsilon^n.