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Convergence of the hypersymplectic flow on T4 with T3‐symmetry

par Fine, Joel ;He, Weiyong;Yao, Chengjian
Référence Journal of the London Mathematical Society, 111, 6
Publication Publié, 2025-06-01

Article révisé par les pairs

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Résumé :

Abstract A hypersymplectic structure on a 4‐manifold is a triple of 2‐forms for which every nontrivial linear combination is a symplectic form. Donaldson has conjectured that when the underlying manifold is compact, any such structure is isotopic in its cohomology class to a hyperkähler triple. We prove this conjecture for a certain class of hypersymplectic structures on which are invariant under the standard action and in what we call “symmetric normal form”. 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 , when starting from a ‐invariant hypersymplectic structure in symmetric normal form, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperkähler structure.