15 juin 2017

Joel Fine (Bruxelles)
Hypersymplectic 4-manifolds and the G2 Laplacian flow

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Lieu : Bâtiment 425, salle 121-123

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.

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

Hypersymplectic 4-manifolds and the G2 Laplacian flow  Version PDF