Estimating Reeb chords using microlocal sheaf theory

TL;DR

This paper establishes lower bounds on the number of Reeb chords for Legendrian submanifolds in 1-jet bundles using microlocal sheaf theory, linking geometric properties to sheaf-theoretic invariants.

Contribution

It introduces a novel approach employing microlocal sheaf theory to estimate Reeb chords, including a duality triangle and persistence structures from action filtration.

Findings

Reeb chord count is bounded below by half the Betti sum of the Legendrian.

Lower bounds for Reeb chords between Legendrian and its pushoff depend on Betti numbers and Hamiltonian oscillation.

Develops a duality exact triangle and uses persistence structures in microlocal sheaves.

Abstract

We show that for a closed Legendrian submanifold in a 1-jet bundle, if there is a sheaf with compact support, perfect stalk and singular support on that Legendrian, then (1) the number of Reeb chords has a lower bound by half of the sum of Betti numbers of the Legendrian; (2) the number of Reeb chords between the original Legendrian and its Hamiltonian pushoff has a lower bound in terms of Betti numbers when the oscillation norm of the Hamiltonian is small comparing with the length of Reeb chords. In the proof we develop a duality exact triangle and use the persistence structure (which comes from the action filtration) of microlocal sheaves.