2023/07/26 by Ronaldo F. de Lima, de Lima, R. F., A. K. Ramos +3 · 3 citations
Mathematics · Physics and Astronomy · Engineering · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Fluid Dynamics and Turbulent Flows
paper · pdf · doi:10.48550/arxiv.2307.14136
We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic 3-space \mathbb H3, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in \mathbb H3, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in \mathbb H3 and apply it to prove that properly immersed translators to MCF in \mathbb 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 x3-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in \mathbb H3 which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in \mathbb 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 \mathbb R3.