Remarks on homogeneous solitons of the $\protect \mathrm{G}_{2}$-Laplacian flow

Anna Fino,Alberto Raffero

doi:10.5802/crmath.39

Abstract

We show the existence of expanding solitons of the G$_2$-Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched G$_2$-structure.

Highlights

A G2-structure on a 7-manifold M is said to be closed if the defining positive 3-form φ ∈ Ω3+(M ) satisfies the equation d φ = 0
The intrinsic torsion of a closed G2-structure can be identified with the unique 2-form τ for which d ∗φ φ = τ ∧ φ = − ∗φ τ, where ∗φ denotes the Hodge operator associated with the Riemannian metric gφ and orientation dVφ induced by φ
A closed G2-structure φ is called a Laplacian soliton if it satisfies the equation

Summary

Introduction

All examples obtained so far are given by seven-dimensional, connected, connected solvable Lie groups G endowed with a left-invariant closed G2-structure φ satisfying (1) with respect to a special vector field X [3, 6, 9,10,11, 15]. We give an example of a connected solvable Lie group endowed with a steady algebraic soliton that is not an extremally Ricci pinched G2-structure. This example satisfies a further remarkable property: there exists a left-invariant vector field X on the Lie group for which ∆φφ = LX φ. This is the first example of a left-invariant closed G2-structure satisfying the equation (1) with respect to a left-invariant vector field

Algebraic Laplacian solitons on Lie groups

Laplacian solitons on non-solvable Lie groups

A steady soliton that is not extremally Ricci pinched