A report on the hypersymplectic flow

Abstract

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and 7-dimensional $G_2$-geometry. We also survey recent progress on the flow, most notably an extension theorem assuming a bound on scalar curvature. The second half contains new results. We prove that a complete torsion-free hypersymplectic structure must be hyperkähler. We show that a certain integral bound involving scalar curvature rules out a finite time singularity in the hypersymplectic flow. We show that if the initial hypersymplectic structure is sufficiently close to being point-wise orthogonal then the flow exists for all time. Finally, we prove convergence of the flow under some strong assumptions including, amongst other things, long time existence.

arXiv comment25 pages. To appear in an issue of PAMQ celebrating Simon Donaldson's 60th birthday. v2 added MSC classes and grant details

References in corpus (5)

• Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle
• Laplacian flow of homogeneous G2-structures and its solitons
• Shi-type estimates and finite time singularities of flows of G${}_2$ structures
• The diversity of symplectic Calabi-Yau six-manifolds
• Local curvature estimates for the Laplacian flow

Cited by in corpus (1)

• Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$