Laplacian solitons on nilpotent Lie groups

TL;DR

This paper studies Laplacian solitons on nilpotent Lie groups with closed G_2-structures, identifying specific Lie algebras that admit such solitons and analyzing their properties and evolution under the flow.

Contribution

It classifies which nilpotent Lie algebras with closed G_2-structures admit Laplacian solitons and explores their continuous families and flow behaviors.

Findings

Seven of twelve Lie algebras admit Laplacian solitons.

One Lie algebra admits a continuous family of non-homothetic solitons.

Flow evolution is non-diagonal in four cases.

Abstract

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton. Moreover, one of them admits a continuous family of Laplacian solitons which are pairwise non-homothetic and the Laplacian flow evolution of four of them is not diagonal.