Bott-integrable Reeb flows on 3-manifolds

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.

arXiv comment36 pages, 5 figures; v2: new Section 9, some references added; v3: new Section 1.3 and other small changes

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• Surgery diagrams for contact 3-manifolds
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Cited by in corpus (1)

• Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems