1 December 2018 Hypersymplectic 4-manifolds, the G2-Laplacian flow, and extension assuming bounded scalar curvature
Joel Fine, Chengjian Yao
Duke Math. J. 167(18): 3533-3589 (1 December 2018). DOI: 10.1215/00127094-2018-0040


A hypersymplectic structure on a 4-manifold X is a triple ω̲ 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, ω̲ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω̲ on a compact manifold X defines a natural G2-structure ϕ on X×T3 which has vanishing torsion precisely when ω̲ 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).


Download Citation

Joel Fine. Chengjian Yao. "Hypersymplectic 4-manifolds, the G2-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


Received: 8 May 2017; Revised: 29 May 2018; Published: 1 December 2018
First available in Project Euclid: 6 November 2018

zbMATH: 07009771
MathSciNet: MR3881202
Digital Object Identifier: 10.1215/00127094-2018-0040

Primary: 53C44
Secondary: 53C26 , 53D35

Keywords: $G_{2}$ manifolds , Geometric flows , hyper-Kähler manifolds , symplectic geometry

Rights: Copyright © 2018 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.167 • No. 18 • 1 December 2018
Back to Top