This paper initiates a classification programme of flows of \(\textrm{SU}(2)\) -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 \(\textrm{G}_2\) 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 \(\textrm{G}_2\) , \(\textrm{Spin}(7)\) and \(\textrm{SU}(3)\) 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 \(\textrm{SU}(2)\) -structure. We compute the first variation formulae of all the quadratic functionals in the torsion; these provide natural building blocks for \(\textrm{SU}(2)\) gradient flows. In particular, our results demonstrate that both the negative gradient flow of the Dirichlet energy of the intrinsic torsion and the harmonic Ricci flow are parabolic after a modified DeTurck’s trick.
