Laplacian solitons: questions and homogeneous examples

TL;DR

This paper presents the first known examples of shrinking closed Laplacian solitons with finite-time singularities, and identifies steady Laplacian solitons with extremally Ricci pinched G2-structures on solvable Lie groups.

Contribution

It provides the first explicit examples of shrinking closed Laplacian solitons and explores steady solitons with special G2-structures on solvable Lie groups.

Findings

First examples of shrinking closed Laplacian solitons.

Existence of steady Laplacian solitons with extremally Ricci pinched G2-structures.

All examples are on solvable Lie groups.

Abstract

We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invariant G2-structures on solvable Lie groups.