Horo-shrinkers in the Hyperbolic Space

A surface $\Sigma$ in the hyperbolic space $\mathbb{H}^{3}$ is called a horo-shrinker if its mean curvature $H$ satisfies $H = \langle N, \partial_{z} \rangle$, where $(x,y,z)$ are the coordinates of $\mathbb{H}^{3}$ in the upper half-space model and $N$ is the unit normal of $\Sigma$. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of $\mathbb{H}^{3}$ 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 along vertical geodesic tangent to $\partial_{z}$. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations perpendicular to $\partial_{z}$. 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 about the $z$-axis, distinguishing if the surfaces intersect or not the rotation axis.