Convergence of the hypersymplectic flow on T4T⁴ with T3T³‐symmetry

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.