The extension of hypersymplectic flow under bounded torsion and convergence of global solution
摘要: A triple of symplectic forms on a differentiable 4-manifold, spanning a maximal positive subspace of Lambda^2 at each point is called a hypersymplectic structure. This notion was introduced by Donaldson in his programme of studying the adiabatic limits of G_2 manifolds. HyperKaehler manifolds of dimension 4 give rich sources of hypersymplectic structures, and were conjectured by Donaldson to be the "only sources" in the compact case. We study a geometric flow---hypersymplectic flow-- of such structures, designed to deform a given hypersymplectic structure to a hyperKaehler one in the same cohomology class. This 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. We show that the hypersymplectic flow does not develop finite time singularity with uniformly bounded torsion. We also study the convergence of the global solution of the flow under various geometric conditions. This is joint work with Joel Fine.
20170808Chengjian Yao The extension of hypersymplectic flow under bounded torsion and convergence of global solution