Conformal solitons for the mean curvature flow in hyperbolic space

Abstract

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space \(\mathbb {H}^{n+1}\). Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field \(-\partial _0\). We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of \(\partial _\infty \mathbb {H}^{n+1}\). We conclude by proving rigidity results for bowl and grim-reaper cylinders.