Flows of SU(2)-structures

This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-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 $\mathrm{G}_2$ 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 $\mathrm{G}_2$, $\mathrm{Spin}(7)$ and $\mathrm{SU}(3)$ 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 $\mathrm{SU}(2)$-structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for $\mathrm{SU}(2)$ 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.