$S^1$-invariant Laplacian flow

TL;DR

This paper studies the $S^1$-invariant Laplacian flow related to $G_2$-structures, deriving evolution equations, discovering new shrinking solitons, and analyzing symmetry properties of solitons.

Contribution

It derives explicit evolution equations for $S^1$-invariant Laplacian flow and presents the first examples of inhomogeneous shrinking solitons, including gradient solitons.

Findings

Derived evolution equations for $S^1$-invariant Laplacian flow.

Discovered the first inhomogeneous shrinking solitons, which are gradient.

Proved that compact non-torsion free solitons have no infinitesimal symmetry.

Abstract

The Laplacian flow is a geometric flow introduced by Bryant as a way for finding torsion free $G_2$-structures. If the flow is $S^1$-invariant then it descends to a flow of $SU(3)$-structures on a $6$-manifold. In this article we derive expressions for these evolution equations. In our search for examples we discover the first inhomogeneous shrinking solitons, which are also gradient. We also show that any compact non-torsion free soliton admits no infinitesimal symmetry.