Harmonic Flow of Quaternion-Kähler Structures

We formulate the gradient Dirichlet flow of Sp(2)Sp(1) -structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G_2 case. We also draw several comparisons with Spin(7) -structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ϵ -regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3) . We also study compactness and the formation of singularities.