A report on the hypersymplectic flow

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 pointwise 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.

2010 Mathematics Subject Classification

53C26, 53C44

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J. Fine was supported by ERC consolidator grant 646649 “SymplecticEinstein”.

Received 8 October 2018

Published 20 March 2020