Horo-shrinkers in the hyperbolic space

A surface $\mathrm{\Sigma }$ in the hyperbolic space ${\text{h}}^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=⟨N,{\mathrm{\partial }}_z⟩$, where $(x,y,z)$ are the coordinates of ${\text{h}}^3$ in the upper half-space model and $N$ is the unit normal of $\mathrm{\Sigma }$. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of ${\text{h}}^3$ depending if these isometries are hyperbolic, parabolic or spherical. We characterize totally geodesic planes as the only horo-shrinkers invariant by a one-parameter group of hyperbolic translations. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations. We describe the geometry of the grim reapers proving that they are periodic surfaces. In the last part of the paper, we give a complete classification of horo-shrinkers invariant by spherical rotations, distinguishing if the surfaces intersect or not the rotation axis.