Reeb flows transverse to foliations

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.

39 pages, 3 figures. v2: a more elementary proof of proposition 2, and several corrections/improvements to the exposition

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