Translators of the mean curvature flow in the special linear group $SL(2,\mathbb{R})$

TL;DR

This paper classifies and explicitly describes surfaces called translators in the special linear group $SL(2,\mathbb{R})$ that move under mean curvature flow, focusing on those invariant under certain symmetry groups.

Contribution

It provides a comprehensive classification of invariant translators in $SL(2,\mathbb{R})$, including explicit parametrizations for some cases, expanding understanding of mean curvature flow in this setting.

Findings

Classified all invariant translators under one-parameter isometry groups.

Derived explicit parametrizations for certain invariant translators.

Analyzed the structure of translators using Iwasawa decomposition.

Abstract

Translators in the special linear group $SL(2,\mathbb{R})$ are surfaces whose mean curvature $H$ and unit normal vector $N$ satisfy $H=\langle N,X\rangle$, where $X$ is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is $4$ and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.