4 may 2017 -- 14:30
Cortona
Abstract.
Hypersymplectic structure was introduced by Donaldson in his programme of studying the adiabatic limits of $G_2$ manifolds. It is triple of symplectic forms on a differentiable $4$-manifold, spanning a maximal positive subspace of $\Lambda^2$ at each point. HyperKaehler manifolds of dimension $4$ give rich sources of hypersymplectic structures, and were conjectured by Donaldson to be the ``only source" in the compact case. We study a geometric flow of such structures, designing to deform a given hypersymplectic structure to a hyperKaehler one in the same cohomology class. We show that the flow does not develop finite time singularity with uniformly bounded torsion. Notice that our flow is a dimensional reduction of the more well-known $G_2$ Laplacian flow introduced by Hitchin and Bryant to study the existence of metrics with $G_2$ holonomy. This is a joint work with Joel Fine.