Horo-shrinkers in the Hyperbolic Space

Abstract

A surface Σ in the hyperbolic space H 3 is called a horo-shrinker if its mean curvature H satisfies H = ⟨ N , ∂ z ⟩ , where ( x , y , z ) are the coordinates of H 3 in the upper half-space model and N is the unit normal of Σ . In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of 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 ∂ z . The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations perpendicular to ∂ 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.