Solitons to mean curvature flow in the hyperbolic 3-space

Le résumé fourni par la source

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} .

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