Bach flow of simply connected nilmanifolds

Abstract

The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the L 2-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger–Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a complete description of the Bach flow on simply connected nilmanifolds.