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

Romina M. Arroyo; Gabriela P. Ovando; Mariel Sáez

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arXiv · 2505.17892

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

Romina M. Arroyo·Gabriela P. Ovando·Mariel Sáez

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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.

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Romina M. Arroyo, Gabriela P. Ovando, Mariel Sáez. 2025-05-26. The mean curvature flow of subgroups on Lie groups of dimension three. https://arxiv.org/abs/2505.17892

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