# Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature

## TL;DR

This paper investigates the deformation of hypersymplectic structures on 4-manifolds using $G_2$-geometry and the $G_2$-Laplacian flow, establishing conditions for flow extension based on scalar curvature bounds.

## Contribution

It links hypersymplectic structures to $G_2$-structures and proves that the $G_2$-Laplacian flow extends as long as scalar curvature remains bounded, providing new insights into their deformation.

## Key findings

- Flow extends with bounded scalar curvature
- Lower bounds for maximal flow existence time
- Connection between hypersymplectic and $G_2$-geometry

## Abstract

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive-definite subspace of $\Lambda^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyperk\"ahler triple. We approach this via a link with $G_2$-geometry. A hypersymplectic structure $\underline\omega$ on a compact manifold $X$ defines a natural $G_2$-structure $\phi$ on $X \times \mathbb{T}^3$ which has vanishing torsion precisely when $\underline{\omega}$ is a hyperk\"ahler triple. We study the $G_2$-Laplacian flow starting from $\phi$, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_2$-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).