Pre-Lagrangian tori transverse to an Anosov flow

TL;DR

This paper investigates the conditions under which a torus transverse to an Anosov flow in a 3-manifold can be associated with a Lagrangian in the Liouville structure on  imes M, revealing topological obstructions and specific positive cases.

Contribution

It identifies topological obstructions preventing certain tori from forming Lagrangians and characterizes when such tori can indeed be associated with Lagrangians based on foliation properties.

Findings

Topological obstructions are linked to foliations induced on the torus.

Lagrangian existence depends on the absence of parallel compact leaves in the foliations.

Positive results are established when the induced foliations lack parallel compact leaves.

Abstract

An Anosov flow on a smooth three-manifold $M$ gives rise to a Liouville structure on $\mathbb{R} \times M$ by a construction of Mitsumatsu. In a recent paper, Cieliebak, Lazarev, Massoni and Moreno ask whether an embedded torus $\Sigma \subseteq M$ transverse to an Anosov flow gives rise to a Lagrangian in $\mathbb{R} \times M$. We show that the answer to this question is in general negative by finding a topological obstruction related to the foliations induced on the torus by the weak stable and unstable bundles of the flow. Going in the opposite direction, we show that the answer is positive if the induced foliations do not admit parallel compact leaves.