The foliated Weinstein conjecture

TL;DR

This paper proves a version of the Weinstein conjecture for foliations with an overtwisted leaf, establishing the existence of Reeb orbits in this context and demonstrating the result's sharpness.

Contribution

It introduces a new result extending the Weinstein conjecture to foliations with overtwisted leaves, a previously unaddressed setting.

Findings

Existence of Reeb orbits in foliations with overtwisted leaves

The result is sharp, meaning it cannot be generally improved

Extends the scope of the Weinstein conjecture to foliated contact structures

Abstract

A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the presence of an overtwisted leaf. The result is shown to be sharp.