A new family of translating solitons in hyperbolic space

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