On homogeneous closed gradient Laplacian solitons

TL;DR

This paper characterizes homogeneous closed gradient Laplacian solitons, demonstrating that many known examples are not gradient except in specific cases, and classifies certain gradient solitons on solvmanifolds.

Contribution

It provides a structure theorem for homogeneous closed gradient Laplacian solitons and shows many known examples are not gradient, except under particular conditions.

Findings

Nicolini's Laplacian solitons on nilpotent Lie groups are not gradient except for N_1 with Gaussian potential.

Fernández-Fino-Manero's closed G_2-structure on N_{12} cannot be a gradient soliton.

Gradient Laplacian solitons on almost abelian solvmanifolds are isometric to products with constant potential functions.

Abstract

We prove a structure theorem for homogeneous closed gradient Laplacian solitons and use it to show some examples of closed Laplacian solitons cannot be made gradient. More specifically, we show that the Laplacian solitons on nilpotent Lie groups found by Nicolini are not gradient up to homothetic $G_2$-structures except for $N_1$, where $f$ must be a Gaussian. We also show that the closed $G_2$-structure $\varphi_{12}$ on $N_{12}$ constructed by Fern\'andez-Fino-Manero cannot be a gradient soliton. We further show that closed non-torsion-free gradient Laplacian solitons on almost abelian solvmanifolds are isometric to products $N \times \mathbb R^k$ with $f$ constant on $N$.