Duke Mathematical Journal

Hypersymplectic 4-manifolds, the G2-Laplacian flow, and extension assuming bounded scalar curvature

Joel Fine and Chengjian Yao

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Abstract

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).

Article information

Source
Duke Math. J., Volume 167, Number 18 (2018), 3533-3589.

Dates
Received: 8 May 2017
Revised: 29 May 2018
First available in Project Euclid: 6 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1541473293

Digital Object Identifier
doi:10.1215/00127094-2018-0040

Mathematical Reviews number (MathSciNet)
MR3881202

Zentralblatt MATH identifier
07009771

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Keywords
symplectic geometry geometric flows $G_{2}$ manifolds hyper-Kähler manifolds

Citation

Fine, Joel; Yao, Chengjian. Hypersymplectic 4-manifolds, the $G_{2}$ -Laplacian flow, and extension assuming bounded scalar curvature. Duke Math. J. 167 (2018), no. 18, 3533--3589. doi:10.1215/00127094-2018-0040. https://projecteuclid.org/euclid.dmj/1541473293


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