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

Abstract

A hypersymplectic structure on a 4-manifold is a triple ${\omega }_1,{\omega }_2,{\omega }_3$ of 2-forms for which every non-trivial linear combination $a^1{\omega }_1+a^2{\omega }_2+a^3{\omega }_3$ 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.

25 pages