Summary: We reduce the embedding problem for hypo \(\mathrm{SU}(2)\) and \(\mathrm{SU}(3)\)-structures to the embedding problem for hypo \(G_2\)-structures into parallel \(\mathrm{Spin7}\)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial \(G_2\)-structure and allows us to prove that the evolution equations, for all of the above embedding problems, do not admit non-trivial longtime solutions. For \(G_2\)-structures we introduce a new flow, which generalizes Hitchin’s flow equations. This intrinsic torsion flow admits unique solutions in the real analytic category.
We extend the Kähler-Ricci flow to \(\mathrm{SU}(n)\)-structures and characterize under which conditions this flow converges to a parallel \(\mathrm{SU}(n)\)-structure. This approach also yields an extension of the Ricci flow to \(G_2\) and \(\mathrm{Spin7}\)-structures. For \(\mathrm{SU}(3)\)-structures in dimension seven we derive the analogue of the Gray-Hervella classification. Based on this classification, we define a type of \(G_2\)-structure which can be regarded as the seven dimensional analogue of Kähler \(\mathrm{SU}(3)\)-structures. This type of \(G_2\)-structures allow a fibrewise Ricci flow that converges to a Ricci flat \(G_2\)-structure.