Reeb chords of Lagrangian slices

TL;DR

This paper proves that the boundary of certain embedded Lagrangian submanifolds in Weinstein domains must have Reeb chords, linking Floer homology properties to boundary dynamics in symplectic geometry.

Contribution

It establishes the necessity of Reeb chords on boundaries of compact exact Lagrangians in subcritical Weinstein domains, connecting Floer homology and boundary deformation obstructions.

Findings

Reeb chords exist on boundaries of embedded Lagrangians in subcritical Weinstein domains.

Obstruction to deforming boundary to a Legendrian cylinder implies Reeb chords.

Vanishing wrapped Floer homology indicates the presence of Reeb chords.

Abstract

In this short note we observe that the boundary of a properly embedded compact exact Lagrangian sub-manifolds in a subcritical Weinstein domain $X$ necessarily admits Reeb chords. The existence of this Reeb chords either follows from an obstruction to the deformation of the boundary to a cylinder over a Legendrian sub-manifold or from the fact that the wrapped Floer homology of the Lagrangian vanishes once this boundary have been "collared".