Hypersymplectic 4-manifolds and the G2 Laplacian flow

Jeudi 15 juin 2017 14:00-15:00 - Joel Fine - Bruxelles

Résumé : A hypersymplectic structure on a 4-manifold is a triple of symplectic forms $w_1$, $w_2$, $w_3$ with the property that at every point $w_i\wedge w_j$ is a positive definite matrix times a volume form. The obvious example is the triple of Kähler forms coming from a hyperkähler metric, where $w_i\wedge w_j$ is the identity matrix times the volume form of the metric. A conjecture of Donaldson states that on a compact 4-manifold and up to isotopy, this is the only possibility : any hypersymplectic structure is isotopic through a path of hypersymplectic structures to a hyperkähler triple. This can be seen as a special case of a folklore conjecture : any symplectic 4-manifold with $c_1=0$ and $b_+=3$ admits a compatible complex structure making it hyperkähler.
I will report on joint work with Chengjian Yao, in which we study a geometric flow designed to deform a given hypersymplectic structure towards a hyperkähler one. The flow comes from a dimensional reduction of G2 geometry. The hypersymplectic structure defines a G2 structure on the product of the 4-manifold with a 3-torus and the G2-Laplacian flow on this 7-manifold determines a flow of hypersymplectic structures on the 4-manifold, called the “hypersymplectic flow”. Our main result is that the hypersymplectic flow exists for as long as the scalar curvature of the 7-manifold remains bounded. One can compare this with the Ricci flow, where the analogous result involves a bound on the whole Ricci curvature.
I will assume no prior knowledge of Ricci flow, G2 geometry or hypersymplectic structures and will do my best to focus on the overall picture rather than technical details.

Lieu : Bâtiment 425, salle 121-123

Notes de dernières minutes : Café culturel assuré à 13h par Hugues Auvray.

Hypersymplectic 4-manifolds and the G2 Laplacian flow  Version PDF