# Invariant translators of the Solvable group

## TL;DR

This paper classifies invariant translators of the mean curvature flow in the solvable group $Sol_3$, revealing new examples of graphical translators and establishing non-existence results, thus advancing understanding of geometric flows in non-Euclidean spaces.

## Contribution

It provides a classification of invariant translators in $Sol_3$, including existence of graphical solutions and non-existence results, which was previously unknown.

## Key findings

- $Sol_3$ admits graphical translators on a half-plane.
- Non-existence results for certain invariant translators.
- Contrast with Euclidean space rigidity results.

## Abstract

We classify the translators to the mean curvature flow in the three-dimensional solvable group $Sol_3$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that $Sol_3$ admits graphical translators defined on a half-plane, in contrast with a rigidity result of Shahriyari for translators in the Euclidean space. Moreover we exhibit some non-existence results.