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arXiv · 2103.01325

Reeb flows transverse to foliations

Jonathan Zung
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Abstract

Let $\mathcal F$ be a co-oriented $C^2$ foliation on a closed, oriented 3-manifold. We show that $T\mathcal F$ can be perturbed to a contact structure with Reeb flow transverse to $\mathcal F$ if and only if $\mathcal F$ 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 $\mathcal F$; this gives a new perspective on the Eliashberg--Thurston theorem.

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Jonathan Zung. 2021-03-01. Reeb flows transverse to foliations. https://doi.org/10.2140/gt.2024.28.3661

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