A hypersymplectic structure on a 4-manifold is a triple of symplectic forms which at every point span a maximal positive definite subspace of for the wedge product. This article is motivated by a conjecture by Donaldson: when is compact, can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with -geometry. A hypersymplectic structure on a compact manifold defines a natural -structure on which has vanishing torsion precisely when is a hyper-Kähler triple. We study the -Laplacian flow starting from , 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 -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).
"Hypersymplectic 4-manifolds, the -Laplacian flow, and extension assuming bounded scalar curvature." Duke Math. J. 167 (18) 3533 - 3589, 1 December 2018. https://doi.org/10.1215/00127094-2018-0040