Bott-integrable Reeb flows on 3-manifolds

TL;DR

This paper investigates Bott integrability of Reeb flows on contact 3-manifolds, showing such flows exist precisely on graph manifolds and analyzing their properties on various specific manifolds.

Contribution

It establishes the existence of Bott-integrable Reeb flows on all contact structures on certain 3-manifolds and develops topological methods relevant to their study.

Findings

Bott-integrable Reeb flows exist exactly on graph manifolds.

All $S^1$-invariant contact structures on Seifert manifolds admit such flows.

Contact structures on $S^3$, $T^3$, and $S^1\times S^2$ admit Bott-integrable Reeb flows.

Abstract

This paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds. We show, in analogy with work of Fomenko-Zieschang on Hamiltonian flows in dimension 4, that Bott-integrable Reeb flows exist precisely on graph manifolds. We also show that all $S^1$-invariant contact structures on Seifert manifolds, as well as all contact structures on the 3-sphere, on the 3-torus, and on $S^1\times S^2$, admit Bott-integrable Reeb flows. Along the way, we establish some general Liouville-type theorems for Bott-integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.