Homothetical surfaces with constant mean curvature in hyperbolic space

We classify all homothetical surfaces with constant mean curvature $H$ in the hyperbolic space $\mathbb{H}^3$. Using the upper half-space model with standard coordinates $(x,y,z)$, these surfaces are defined by the relation $z = \phi(x)\psi(y)$, where $\phi$ and $\psi$ are smooth functions of one variable. We demonstrate that any such surface is necessarily parabolic, meaning that either $\phi$ or $\psi$ is a constant function. Our results cover the minimal case ($H=0$), the case $H^2 \neq 1$, and the critical case $H^2=1$, thereby extending the existing classification of parabolic surfaces in hyperbolic space.