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

We investigate left-invariant Hitchin and hypo flows on five-, six- and seven-dimensional Lie groups. They provide Riemannian cohomogeneity-1 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 1 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-1 Riemannian metric with holonomy equal to G(2) on the twisted product SL(2, C) xSU(2) C-2 described by Bryant and Salamon.