A hypersymplectic structure on a 4-manifold $X$ is a triple$underline{omega}$ of symplectic forms which at every point span a maximalpositive-definite subspace of $Lambda^2$ for the wedge product. This articleis motivated by a conjecture of Donaldson: when $X$ is compact$underline{omega}$ can be deformed through cohomologous hypersymplecticstructures to a hyperk"ahler triple. We approach this via a link with$G_2$-geometry. A hypersymplectic structure $underlineomega$ on a compactmanifold $X$ defines a natural $G_2$-structure $phi$ on $X imesmathbb{T}^3$ 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 mainresult is that the flow extends as long as the scalar curvature of thecorresponding $G_2$-structure remains bounded. An application of our result isa lower bound for the maximal existence time of the flow, in terms of weakbounds on the initial data (and with no assumption that scalar curvature isbounded along the flow).
展开▼
