Convergence of the hypersymplectic flow on $T^4$ with $T^3$-symmetry

Joel Fine; Weiyong He; Chengjian Yao

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arXiv · 2404.15016

Convergence of the hypersymplectic flow on $T^4$ with $T^3$-symmetry

Joel Fine·Weiyong He·Chengjian Yao

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Abstract

A hypersymplectic structure on a 4-manifold is a triple $ω_1, ω_2, ω_3$ of 2-forms for which every non-trivial linear combination $a^1ω_1 + a^2 ω_2 + a^3 ω_3$ is a symplectic form. Donaldson has conjectured that when the underlying manifold is compact, any such structure is isotopic in its cohomolgy class to a hyperkähler triple. We prove this conjecture for a hypersymplectic structure on $T^4$ which is invariant under the standard $T^3$ action. The proof uses the hypersymplectic flow, a geometric flow which attempts to deform a given hypersymplectic structure to a hyperkähler triple. We prove that on $T^4$, when starting from a $T^3$-invariant hypersymplectic structure, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperkähler structure.

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BibTeX RIS

Joel Fine, Weiyong He, Chengjian Yao. 2024-04-23. Convergence of the hypersymplectic flow on $T^4$ with $T^3$-symmetry. https://arxiv.org/abs/2404.15016

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