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

TL;DR

This paper studies the boundary behavior of Hitchin and hypo flows on Lie groups, showing that many such manifolds are incomplete and classifying certain structures on SL(2,C) that relate to known G2 metrics.

Contribution

It proves non-existence of complete cohomogeneity-one manifolds with special holonomy on solvable Lie groups and classifies bi-invariant half-flat structures on SL(2,C) related to G2 metrics.

Findings

Many Hitchin and hypo flow manifolds are incomplete.

Complete manifolds with these structures are flat and have no singular orbits.

Classified bi-invariant half-flat structures on SL(2,C) leading to known G2 metrics.

Abstract

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.