Uniform foliations with Reeb components (Q6065490)

Consider a foliation \(\mathcal{F}\) on a compact Riemannian 3-manifold \(M\), and let the foliation \(\tilde{\mathcal{F}}\) be the lift of \(\mathcal{F}\) to the universal cover \(\tilde{M}\) of \(M\). Then the foliation \(\mathcal{F}\) is said to be \emph{uniform} if any pair of leaves of \(\tilde{\mathcal{F}}\) are at finite Hausdorff distance from each other. Much is known about a uniform foliation \(\mathcal{F}\) if one assumes it to be Reebless. Inspired amongst others by work in this case of \textit{S. R. Fenley} and \textit{R. Potrie} [Groups Geom. Dyn. 15, No. 4, 1489--1521 (2021; Zbl 1490.53039)], the author in the present paper focuses on the opposite case and studies uniform foliations with Reeb components. Answering amongst others a question posed by Fenley and Potrie, the author gives examples of such foliations on a family of closed 3-manifolds with infinite fundamental group. Furthermore, the paper contains some results concerning the behavior of a uniform foliation with Reeb components on general 3-manifolds.