An energy-capacity inequality for Legendrian submanifolds

TL;DR

This paper establishes a lower bound on the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off, linking contact topology with Floer homology techniques.

Contribution

It introduces an energy-capacity inequality relating Reeb chords and Betti numbers, using a Mayer--Vietoris sequence for Lagrangian Floer homology with neck-stretching.

Findings

Number of Reeb chords ≥ sum of $Z_2$-Betti numbers of the submanifold

Invariance under certain contact isotopies with controlled size

Application of Mayer--Vietoris sequence in Floer homology

Abstract

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer--Vietoris sequence for Lagrangian Floer homology, obtained by "neck-stretching" and "splashing."