The mean curvature flow of subgroups on Lie groups of dimension three

TL;DR

This paper investigates the evolution of two-dimensional Lie subgroups under mean curvature flow within three-dimensional Lie groups, highlighting differences between unimodular and non-unimodular cases.

Contribution

It characterizes the mean curvature flow of Lie subgroups in 3D Lie groups, especially focusing on non-unimodular groups where non-trivial evolutions occur.

Findings

Unimodular Lie groups have minimal Lie subgroups under mean curvature flow.

Non-unimodular groups exhibit non-self-similar evolution of Lie subgroups.

Abelian subgroups evolve by isometries, showing self-similarity.

Abstract

In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.