Laplacian Solitons and Symmetry in G_2-geometry

TL;DR

This paper investigates Laplacian solitons in G_2-geometry on compact 7-manifolds, showing they can only be shrinking or steady, and explores the symmetry structure of torsion-free G_2-structures.

Contribution

It proves that Laplacian solitons on compact G_2-manifolds are limited to shrinking or steady types and characterizes their symmetry groups in terms of cohomology.

Findings

Laplacian solitons are only shrinking or steady on compact G_2-manifolds.

The symmetry space of torsion-free G_2-structures is isomorphic to first cohomology.

Comparison with Ricci solitons provides insights into geometric flows.

Abstract

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the space of symmetries (vector fields that annihilate via the Lie derivative) of a torsion-free $G_2$-structure on a compact 7-manifold is canonically isomorphic to $H^1(M,\mathbb{R})$. Some comparisons with Ricci solitons are also discussed, along with some future directions of exploration.