Hypersymplectic 4-Manifolds, The G2-laplacian flow, and extension assuming bounded scalar curvature

Abstract

A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive definite subspace of Λ2 for the wedge product. This article is motivated by a conjecture by Donaldson: when X is compact, ω can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω on a compact manifold X defines a natural G2-structure ϕ on X × T(double-struck)3 which has vanishing torsion precisely when ω is a hyper-Kähler triple.We study the G2-Laplacian flow starting from ϕ, 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 G2-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).