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arXiv · 2302.07701

Bott-integrable Reeb flows on 3-manifolds

Hansjörg Geiges·Jakob Hedicke·Murat Sağlam
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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.

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Hansjörg Geiges, Jakob Hedicke, Murat Sağlam. 2023-12-04. Bott-integrable Reeb flows on 3-manifolds. https://doi.org/10.1112/jlms.12859

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