On Hopf hypersurfaces of the complex hyperbolic quadric with constant principal curvatures

In this paper, we study the Hopf hypersurfaces of the complex hyperbolic quadric $Q^{m\ast }=S{O}_{2,m}^o\mathrm{/}(S{O}_2\times S{O}_m)$ ($m\ge 3$) with constant principal curvatures. We classify the Hopf hypersurfaces of $Q^{m\ast }$ ($m\ge 3$) with at most two distinct constant principal curvatures. For Hopf hypersurfaces with three or four distinct constant principal curvatures, we determine the values of the principal curvatures as well as their multiplicities.