Reeb flows transverse to foliations

TL;DR

This paper establishes conditions under which a Reeb flow can be made transverse to a foliation on a 3-manifold, linking the existence of invariant measures to the ability to perturb the foliation into a contact structure.

Contribution

It provides a new criterion for the existence of transverse Reeb flows, connecting foliation invariants with contact topology, and introduces leafwise Brownian motion as a technical tool.

Findings

Reeb flows can be made transverse if and only if no invariant transverse measure exists.

The constructed Reeb flows have no contractible orbits.

The approach offers a new perspective on the Eliashberg--Thurston theorem.

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.