Flows of SU(2)-structures

This paper initiates a classification programme of flows of SU(2)-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the context of G₂ geometry. We show how this strategy can also be used to deduce the number of geometric flows of a given H-structure; we illustrate this in the G₂, Spin(7) and SU(3) cases. Our investigation also leads us to derive explicit expressions for the Ricci and self-dual Weyl curvature in terms of the intrinsic torsion of the underlying SU(2)-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for SU(2) gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the Ricci harmonic flow are parabolic after a modified DeTurck's trick.