Amanda Petcu, University of Waterloo
Some results on hypersymplectic structures
A conjecture of Simon Donaldson is that on a compact 4-manifold X^4 one can flow from a hypersymplecticstructure to a hyperkahler structure while remaining in the same cohomology class. To this end thehypersymplectic flow was introduced by Fine-Yao. In this thesis the notion of a positive triple on X^4 is used todefine a hypersymplectic and hyperkahler structure. Given a closed positive triple one can define either a closedG2 structure or a coclosed G2 structure on T^3 x X^4. The coclosed G2 structure is evolved under the G2Laplacian coflow. This descends to a flow of the positive triple on X^4, which is again the Fine-Yaohypersymplectic flow. In the second part of this thesis we let X^4 = R^4 \0 with a particular cohomogeneity oneaction. A hypersymplectic structure invariant under this action is introduced. The Riemann and Ricci curvaturetensors are computed and we verify in a particular case that this hypersymplectic structure can be transformed toa hyperkahler structure. The notion of a soliton for the hypersymplectic flow in this particular case is introducedand it is found that steady solitons give rise to hypersymplectic structures that can be transformed to hyperkahlerstructures. Some other soliton solutions are also discussed.
MC 5403