The stable Morse number as a lower bound for the number of Reeb chords

TL;DR

This paper establishes a new lower bound for the number of Reeb chords on Legendrian submanifolds using the stable Morse number of their Lagrangian fillings, improving previous bounds based on Betti numbers.

Contribution

It introduces a novel lower bound for Reeb chords based on the stable Morse number, extending prior bounds that relied on Betti numbers.

Findings

Reeb chord count is bounded below by the stable Morse number of the filling.

The bound depends on the homotopy type of the Lagrangian filling.

Improves previous bounds based on Betti numbers.

Abstract

Assume that we are given a closed chord-generic Legendrian submanifold $\Lambda \subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $\Lambda$ moreover admits an exact Lagrangian filling $L_{\Lambda} \subset \mathbb R \times P \times \mathbb R$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $\Lambda$ is bounded from below by the stable Morse number of $L_{\Lambda}$. Given a general exact Lagrangian filling $L_{\Lambda}$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_{\Lambda}$, following Ono-Pajitnov's implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $\Lambda$ or $L_{\Lambda}$.