Reeb flows transverse to foliations

Zung, Jonathan¹

¹ Department of Mathematics, Princeton University, Princeton, NJ, United States

Geometry & topology, Tome 28 (2024) no. 8, pp. 3661-3695.

Résumé

Let $\mathcal{ℱ}$ be a cooriented $C^2$ 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.

Détail

DOI : 10.2140/gt.2024.28.3661

Keywords: Reeb flow, contact structure, foliation

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