Bach Flow on Homogeneous Products

TL;DR

This paper analyzes the qualitative behavior of Bach flow on four-dimensional locally homogeneous product manifolds, providing explicit solutions and understanding the limiting behavior of metrics and curvature.

Contribution

It establishes the behavior of Bach flow on these manifolds by analyzing differential equations and explicitly solving in certain cases, especially on quotients of  imes .

Findings

Explicit solutions for Bach flow in some cases.

Determination of limiting behavior of metrics and curvature.

Analysis of complex cases like quotients of  imes .

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.