Solitons to mean curvature flow in the hyperbolic 3-space

Abstract

We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space ${\mathbb{H}}^3$, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete translators in ${\mathbb{H}}^3$, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in ${\mathbb{H}}^3$ and apply it to prove that properly immersed translators to MCF in ${\mathbb{H}}^3$ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the $z$-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in ${\mathbb{H}}^3$ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in ${\mathbb{H}}^3$ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in ${\mathbb{R}}^3$.