The class of grim reapers in ${\mathbb{H}}^2\times \mathbb{R}$

Abstract

We study translators of the mean curvature flow in the product space $\h^2\times\r$. In $\h^2\times\r$ there are three types of translations: vertical translations due to the factor $\r$ and parabolic and hyperbolic translations from $\h^2$. A grim reaper in $\h^2\times\r$ is a translator invariant by a one-parameter group of translations. The variety of translators and translations in $\h^2\times\r$ makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of $\h^2\times\r$ with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

22 pages, 8 figures. Last section has been improved, updated bibliography and new figures have been added