Convergence of the hypersymplectic flow on T⁴ with T³-symmetry

A hypersymplectic structure on a 4-manifold is a triple ω₁, ω₂, ω₃ of 2-forms for which every non-trivial linear combination a¹ω₁ + a² ω₂ + a³ ω₃ 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 hyperk\"ahler triple. We prove this conjecture for a hypersymplectic structure on T⁴ which is invariant under the standard T³ action. The proof uses the hypersymplectic flow, a geometric flow which attempts to deform a given hypersymplectic structure to a hyperk\"ahler triple. We prove that on T⁴, when starting from a T³-invariant hypersymplectic structure, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperk\"ahler structure.