Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 027, 35 pages      arXiv:1803.07733      https://doi.org/10.3842/SIGMA.2020.027

Bach Flow on Homogeneous Products

Dylan Helliwell
Department of Mathematics, Seattle University, 901 12th Ave, Seattle, WA 98122, USA

Received September 03, 2019, in final form March 29, 2020; Published online April 11, 2020

Abstract
Qualitative behavior of Bach flow is established on compact four-dimensional locally homogeneous product manifolds. This is achieved by lifting to the homogeneous universal cover and, in most cases, capitalizing on the resultant group structure. The resulting system of ordinary differential equations is carefully analyzed on a case-by-case basis, with explicit solutions found in some cases. Limiting behavior of the metric and the curvature are determined in all cases. The behavior on quotients of $\mathbb{R} \times \mathbb{S}^3$ proves to be the most challenging and interesting.

Key words: high-order geometric flows; Bach flow; locally homogeneous manifold; three-dimensional Lie group.

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