A new family of translating solitons in hyperbolic space

If ξ is a Killing vector field of the hyperbolic space ^3 whose flow are parabolic isometries, a surface Σ⊂^3 is a ξ-translator if its mean curvature H satisfies H=⟨ N,ξ⟩, where N is the unit normal of Σ. We classify all ξ-translators invariant by a one-parameter group of rotations of ^3, exhibiting the existence of a new family of grim reapers. We use these grim reapers to prove the non-existence of closed ξ-translators.