On the boundary behaviour of left-invariant Hitchin and hypo flows

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$, respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups $G$ these manifolds cannot be completed: a complete Riemannian manifold with parallel $SU(3)$-, $G_2$- or $Spin(7)$-structure which is of cohomogeneity one with respect to $G$ is flat, and has no singular orbits. We furthermore classify, on the non-compact Lie group $SL(2,C)$, all half-flat $SU(3)$-structures which are bi-invariant with respect to the maximal compact subgroup $SU(2)$ and solve the Hitchin flow for these initial values. It turns out that often the flow collapses to a smooth manifold in one direction. In this way we recover an incomplete cohomogeneity-one Riemannian metric with holonomy equal to $G_2$ on the twisted product $SL(2,C)\times_{SU(2)} C^2$ described by Bryant and Salamon.