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In this paper we study conformal solitons for the mean curvature flow inhyperbolic space $\mathbb{H}^{n+1}$. Working in the upper half-space model, wefocus on horo-expanders, which relate to the conformal field $-\partial_0$. Weclassify cylindrical and rotationally symmetric examples, finding appropriateanalogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, weaddress the Plateau and the Dirichlet problems at infinity. For the latter, weprovide the sharp boundary convexity condition to guarantee its solvability,and address the case of noncompact boundaries contained between two parallelhyperplanes of $\partial_{\infty}\mathbb{H}^{n+1}$. We conclude by provingrigidity results for bowl and grim-reaper cylinders.

Conformal solitons for the mean curvature flow in hyperbolic space