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

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

TL;DR

This work analyzes the mean curvature flow (MCF) for embeddings of two-dimensional Lie subgroups inside three-dimensional Lie groups $G$ endowed with a fixed left-invariant metric, focusing on the distinction between unimodular and non-unimodular ambient groups. By computing the mean curvature via $H = -\mathrm{tr}\,\mathrm{ad}_{\nu}$ and exploiting Killing fields to generate one-parameter isometries, the authors classify all possible 2D subgroups in the relevant non-unimodular 3D groups and identify when their MCF evolutions are translators (self-similar) versus non-self-similar with constant mean curvature. They provide explicit evolutions for abelian subgroups (translating solitons) and for non-abelian subgroups (constant $H$ with explicit, non-self-similar flows), organized by group family $S_{3,\lambda}$, $S_{3,\lambda_1,\lambda_2}$, and $S'_{3,\lambda}$, and deduce concrete translating and non-translating solutions. A counterexample in four dimensions shows limits of extending the 3D translational behavior to higher dimensions. Overall, the paper clarifies how the algebraic structure of the ambient Lie group governs MCF behavior, offering explicit solitons and evolution formulas that may inform higher-dimensional analogues and moduli questions for solvmanifolds.

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.

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

TL;DR

This work analyzes the mean curvature flow (MCF) for embeddings of two-dimensional Lie subgroups inside three-dimensional Lie groups endowed with a fixed left-invariant metric, focusing on the distinction between unimodular and non-unimodular ambient groups. By computing the mean curvature via and exploiting Killing fields to generate one-parameter isometries, the authors classify all possible 2D subgroups in the relevant non-unimodular 3D groups and identify when their MCF evolutions are translators (self-similar) versus non-self-similar with constant mean curvature. They provide explicit evolutions for abelian subgroups (translating solitons) and for non-abelian subgroups (constant with explicit, non-self-similar flows), organized by group family , , and , and deduce concrete translating and non-translating solutions. A counterexample in four dimensions shows limits of extending the 3D translational behavior to higher dimensions. Overall, the paper clarifies how the algebraic structure of the ambient Lie group governs MCF behavior, offering explicit solitons and evolution formulas that may inform higher-dimensional analogues and moduli questions for solvmanifolds.

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.

Paper Structure

This paper contains 23 sections, 9 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Lie groups of dimension three
  3. Three dimensional Lie groups with codimension one subgroups
  4. Subgroups of the Heisenberg Lie group $S_{3,0}$.
  5. Subgroups of $S_{3,\lambda}$.
  6. Subgroups of $S_{3,\lambda_1,\lambda_2}$.
  7. Subgroups of $S_{3,\lambda}'$.
  8. Subgroups of $\mathrm{SL}(2,\mathbb R)$ and $\mathrm{SO}(3)$.
  9. Solitons among Lie subgroups in dimension three
  10. The Lie group $S_{3,\lambda}$
  11. The subgroup $\mathcal{K}_0$
  12. The subgroup $\mathcal{K}_a$
  13. The group $S_{3,\lambda_1,\lambda_2}$
  14. The case $\lambda_1\ne \lambda_2$ and $\lambda_2\ne 0$.
  15. The case $\lambda_2=0$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be a non-unimodular Lie group of dimension three equipped with a fixed left-invariant metric. Let $K$ denote an abelian subgroup of $G$ with non-vanishing mean curvature, then its mean curvature is constant and there is a family of translations $\varphi_t$ such that $\varphi_t(K)$ is a solut

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 3.1
  • Theorem 3.2
  • ...and 1 more