HYPERSYMPLECTIC 4-MANIFOLDS, THE G(2)-LAPLACIAN FLOW, AND EXTENSION ASSUMING BOUNDED SCALAR CURVATURE

摘要

A hypersymplectic structure on a 4-manifold X is a triple (omega) under bar 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 by Donaldson: when X is compact, (omega) under bar can be deformed through cohomologous hypersymplectic structures to a hyper-Kahler triple. We approach this via a link with G(2)-geometry. A hypersymplectic structure (omega) under bar on a compact manifold X defines a natural G(2)-structure phi on X x T-3 which has vanishing torsion precisely when (omega) under bar is a hyper-Kahler 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).