Reeb flows transverse to foliations

Zung, Jonathan

doi:10.2140/gt.2024.28.3661

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Reeb flows transverse to foliations

Jonathan Zung

Geometry & Topology 28 (2024) 3661–3695

DOI: 10.2140/gt.2024.28.3661

Abstract

Let $ℱ$ be a cooriented $C^{2}$ foliation on a closed, oriented $3$ –manifold. We show that $T ℱ$ can be perturbed to a contact structure with Reeb flow transverse to $ℱ$ if and only if $ℱ$ does not support an invariant transverse measure. The resulting Reeb flow has no contractible orbits. This answers a question of Colin and Honda. The main technical tool in our proof is leafwise Brownian motion which we use to construct good transverse measures for $ℱ$ ; this gives a new perspective on the Eliashberg–Thurston theorem.

Keywords

Reeb flow, contact structure, foliation

Mathematical Subject Classification

Primary: 57K33, 57R30

References

Publication

Received: 4 August 2021

Revised: 29 January 2023

Accepted: 16 September 2023

Published: 20 December 2024

Proposed: Yakov Eliashberg

Seconded: David Gabai, David Fisher

Authors

Jonathan Zung
Department of Mathematics Princeton University Princeton, NJ United States

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Volume 28, issue 8 (2024)