# Translating solitons of the mean curvature flow in the space $\mathbb{H}^2\times\mathbb{R}$

## TL;DR

This paper investigates self-translating solitons of the mean curvature flow in hyperbolic space times a line, relating them to minimal surfaces in conformal metrics, providing explicit examples, and establishing uniqueness and non-existence results.

## Contribution

It introduces a novel approach by linking translating solitons to minimal surfaces in a conformal metric space, and constructs explicit examples in 22.

## Key findings

- Explicit examples of translating solitons are constructed.
- Asymptotic behavior of rotationally symmetric solutions is analyzed.
- Uniqueness and non-existence theorems are established.

## Abstract

In this paper we study the theory of self translating solitons of the mean curvature flow of immersed surfaces in the product space $\mathbb{H}^2\times\mathbb{R}$. We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a conformal metric space. Explicit examples of these surfaces are constructed, and we study the asymptotic behaviour of the existing rotationally symmetric examples. Finally, we prove some uniqueness and non-existence theorems.