Laplacian flow of homogeneous G2-structures and its solitons

TL;DR

This paper studies homogeneous Laplacian G2-structures and their solitons, proving their algebraic nature, providing examples, and analyzing long-term behavior and classifications within specific Lie group contexts.

Contribution

It characterizes homogeneous Laplacian solitons as semi-algebraic, distinguishes them from Ricci solitons, and provides explicit examples and classifications in certain Lie group settings.

Findings

Homogeneous Laplacian solitons are equivalent to semi-algebraic solitons.

Existence of a semi-algebraic soliton not equivalent to any algebraic soliton.

Long-term existence and convergence properties of solutions on certain Lie groups.

Abstract

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.