Solitons to Mean Curvature Flow in the hyperbolic 3-space

TL;DR

This paper studies solitons to mean curvature flow in hyperbolic 3-space, establishing existence, uniqueness, and classification results for translators and rotators, including special types like catenoid, cylindrical, and horosphere-based solutions.

Contribution

It provides the first classification of translators and rotators in hyperbolic 3-space, including existence, uniqueness, and properties of these solutions, extending known Euclidean results.

Findings

Existence of two distinct families of complete rotational translators in H^3.

Properly immersed translators in H^3 are not cylindrically bounded.

Complete horoconvex translators are subsets of horospheres.

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 rotational 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 $x_3$-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 $\mathbb R^3$.