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 H³ , providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete translators in H³ , one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in H³ and apply it to prove that properly immersed translators to MCF in H³ 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 H³ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in H³ 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 R³.