Homothetical surfaces with constant mean curvature in hyperbolic space

Abstract

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 $z=\varphi (x)\psi (y)$, where $\varphi$ and $\psi$ are smooth functions of one variable. We demonstrate that any such surface is necessarily parabolic, meaning that either $\varphi$ or $\psi$ is a constant function. Our results cover the minimal case ($H=0$), the case $H^2\ne 1$, and the critical case $H^2=1$, thereby extending the existing classification of parabolic surfaces in hyperbolic space.

22 pages, 1 figure