Hypersurfaces of ${\mathbb{S}}^3\times \mathbb{R}$ and ${\mathbb{H}}^3\times \mathbb{R}$ with constant principal curvatures

We classify the hypersurfaces of ${\mathbb{Q}}^3\times \mathbb{R}$ with three distinct constant principal curvatures, where $\epsilon \in \{1,-1\}$ and ${\mathbb{Q}}^3$ denotes the unit sphere ${\mathbb{S}}^3$ if $\epsilon =1$, whereas it denotes the hyperbolic space ${\mathbb{H}}^3$ if $\epsilon =-1$. We show that they are cylinders over isoparametric surfaces in ${\mathbb{Q}}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of ${\mathbb{Q}}^3\times \mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in ${\mathbb{Q}}^3\times \mathbb{R}$.