Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

TL;DR

This paper introduces an Aleksandrov reflection method for analyzing expanding curvature flows in hyperbolic space, demonstrating convergence to special geometric structures with exponential rates.

Contribution

It develops a new reflection framework applicable to level-set formulations of curvature flows in hyperbolic spaces, establishing convergence and geometric estimates.

Findings

Solutions become starshaped and converge exponentially to an umbilic hypersurface.

Flow solutions in non-compact settings converge to a horosphere with uniform gradient bounds.

The method applies to inverse mean curvature flow as a model case.

Abstract

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. As a consequence, we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become starshaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend our results to the non-compact setting, assuming that the solution has a unique point at infinity. In this case, we prove that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere.