Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.