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

TL;DR

This paper proves that on a 4-torus with a $T^3$-invariant hypersymplectic structure, the hypersymplectic flow exists globally and converges to a hyperk"ahler structure, confirming Donaldson's conjecture in this setting.

Contribution

It demonstrates the long-time existence and convergence of the hypersymplectic flow on $T^4$ with $T^3$ symmetry, confirming Donaldson's conjecture for this case.

Findings

Flow exists for all time on $T^4$ with $T^3$ symmetry.

Flow converges to a hyperk"ahler structure.

Unique cohomologous hyperk"ahler structure is obtained.

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 hyperk\"ahler 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 hyperk\"ahler 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 hyperk\"ahler structure.