Bach Flow of Simply Connected Nilmanifolds

TL;DR

This paper investigates the long-term behavior of the Bach flow on four-dimensional simply connected nilmanifolds, showing convergence to an expanding Bach soliton and providing a complete classification in this setting.

Contribution

It demonstrates the existence, long-time behavior, and convergence of the Bach flow on indecomposable nilmanifolds, extending the understanding of geometric flows on these spaces.

Findings

Bach flow exists for all positive times on these manifolds.

Rescaled flow converges to a non-gradient expanding Bach soliton.

Provides a complete description of Bach flow on simply connected nilmanifolds.

Abstract

The Bach flow is a fourth order geometric flow defined on four manifolds. For a compact manifold, it is a conformally modified 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.