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.

在本文中，我们研究双曲空间\(\mathbb {H}^{n+1}\)中平均曲率流的共形孤子。在上半空间模型中，我们关注 horo 扩展器，它与共形场\(-\partial _0\)相关。我们对圆柱形和旋转对称的例子进行分类，找到死神圆柱体、碗状孤子和翼状孤子的适当类似物。此外，我们还解决了无穷远处的高原问题和狄利克雷问题。对于后者，我们提供尖锐边界凸性条件来保证其可解性，并解决两个平行超平面之间包含非紧边界的情况\(\partial _\infty \mathbb {H}^{n+1}\)。最后，我们证明了碗状气缸和死神气缸的刚性结果。