Some results on hypersymplectic structures

Abstract

A conjecture of Simon Donaldson is that on a compact 4-manifold X⁴ one can flow from a hypersymplectic structure to a hyperkähler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine–Yao. In this thesis the notion of a positive triple on X⁴ is used to define a hypersymplectic and hyperkähler structure. Given a closed positive triple one can define either a closed G₂ structure or a coclosed G₂ structure on 𝕋³ × X⁴. The coclosed G₂ structure is evolved under the G₂ Laplacian coflow. The coflow descends to a flow of the positive triple on X⁴, which is again the Fine–Yao hypersymplectic flow. In the second part of this thesis we let X⁴ = ℝ⁴ ∖ {0} with a particular cohomogeneity one action. A hypersymplectic structure built from data invariant under this action is introduced. The Riemann and Ricci curvature tensors are computed and we verify in a particular case that this hypersymplectic structure can be transformed to a hyperkähler structure. The notion of a soliton for the hypersymplectic flow in this particular case is introduced and it is found that steady solitons give rise to hypersymplectic structures that can be transformed to hyperkähler structures. Some other soliton solutions are also discussed.