Abstract

A hypersymplectic structure on a 4-manifold is a triple w_1, w_2, w_3 of symplectic forms such that any non-zero linear combination of these forms is again symplectic. The prototypical example is a the triple of Kähler forms of a hyperkähler metric. Donaldson has conjectured that up to isotopy, this is the only example. More precisely, Donaldson conjectures that on a compact 4-manifold, any hypersymplectic triple is isotopic through cohomologous hypersymplectic triples to a hyperkähler triple. This is a special case of a famous folklore conjecture: a compact symplectic 4-manifold with c_1=0 and b_+=3 admits a compatible integrable complex structure making it hyperkähler. I will describe an approach to Donaldson’s conjecture which goes via G2 geometry. It gives a natural flow of hypersymplectic structures which tries to deform a given triple into a hyperkähler one. It can be thought of as an analogue of Ricci flow adapted to this context. I will then explain joint work with Chengjian Yao, which shows that the hypersymplectic flow exists as long as the scalar curvature of the associated G2 metrics remains bounded. It is intriguing that this is a stronger existence result than what is currently known for Ricci flow. I will not assume any prior knowledge of Ricci flow, or G2 geometry.