Flows of G2 structures

A G2-structure is a special type of 3-form on an oriented 7-manifold, which determines a Riemannian metric in a nonlinear way. The best class of such 3-forms are those which are parallel with respect to their induced Levi-Civita connections, which is a fully non-linear PDE. More generally, the torsion of a G2-structure is a 2-tensor which quantifies the failure of a G2-structure to be parallel. It is natural to consider geometric flows of G2-structures as a means of starting with a G2-structure with torsion and (hopefully) improving it in some way along the flow. I will begin with an introduction to all of these ideas, and  try to survey some of the results in the field. Then I will talk about work in progress with Dwivedi and Gianniotis to study a large class of flows of G2-structures. In particular, we explicitly describe all possible second order differential invariants of a G2-structure which can be used to construct a quasi-linear second order flow. Then we find conditions on a subclass of these general flows which are amenable to the deTurck trick for establishing short-time existence and uniqueness.