Homothetical surfaces with constant mean curvature in hyperbolic space

TL;DR

This paper classifies all homothetical surfaces with constant mean curvature in hyperbolic space, showing they are necessarily parabolic with either $ ext{phi}$ or $ ext{psi}$ constant, extending previous classifications.

Contribution

It provides a complete classification of homothetical surfaces with constant mean curvature in hyperbolic space, including minimal and critical cases.

Findings

Any such surface is necessarily parabolic.

Either $ ext{phi}$ or $ ext{psi}$ must be constant.

The classification includes minimal, non-zero, and critical mean curvature cases.

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 $z = \phi(x)\psi(y)$, 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.