Harmonic Sp(2)-Invariant G2-Structures on the 7-Sphere

We describe the 10-dimensional space of Sp(2)-invariant G(2)-structures on the homogeneous 7-sphere S-7 = Sp(2)/Sp(1) as Omega(3)(+)(S-7)(sp(2)) similar or equal to R+ x Gl(+)(3, R). In those terms, we formulate a general Ansatz for G2-structures, which realises representatives in each of the 7 possible isometric classes of homogeneous G(2)-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an S 3 -family, the equator of which is a new S-2-family of coclosed G(2)-structures satisfying the harmonicity condition div T = 0. We show general existence of harmonic representatives of G(2)-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic G(2)-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.