New examples of shrinking Laplacian solitons

TL;DR

This paper introduces a new family of shrinking Laplacian solitons on Lie groups, providing the second known solutions to the closed G_2-Laplacian flow with finite-time singularities, and analyzes their torsion and Laplacian operators.

Contribution

It presents a one-parameter family of shrinking Laplacian solitons and studies their torsion forms and Laplacian operators, expanding the known solutions to the closed G_2-Laplacian flow.

Findings

Constructed a new family of shrinking Laplacian solitons.

Proved the non-existence of closed eigenforms under certain conditions.

Analyzed torsion and Laplacian operators on these structures.

Abstract

We give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed $G_2$-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of $G_2$-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family.