The hypersymplectic flow descended from the $G_2$-Laplacian coflow

TL;DR

This paper studies a flow on 4-manifolds that aims to transform hypersymplectic structures into hyperkähler structures, using a modified $G_2$-Laplacian coflow on associated $G_2$ structures.

Contribution

It connects the hypersymplectic flow to a $G_2$-Laplacian coflow on 7-manifolds, providing a new perspective on Donaldson's conjecture.

Findings

The flow is well-defined on positive triples on 4-manifolds.

The coflow induces a flow on the hypersymplectic structures.

The approach links hypersymplectic and $G_2$ geometries.

Abstract

A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperk\"ahler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine-Yao. In this paper the notion of a positive triple on $X^4$ is used to describe a hypersymplectic and hyperk\"ahler structure. Given a closed positive triple one can define either a closed $G_2$ structure or a coclosed $G_2$ structure on $\mathbb{T}^3 \times X^4$. The coclosed $G_2$ structure is evolved under the modified $G_2$-Laplacian coflow. The coflow descends to a flow of the positive triple on $X^4$, which is again the Fine-Yao hypersymplectic flow.