Spectral selectors and contact orderability

TL;DR

This paper establishes a link between orderability of Legendrian submanifolds and spectral selectors, leading to new insights into Reeb chords, contactomorphism spaces, and several longstanding conjectures in contact topology.

Contribution

It proves that orderability is equivalent to the existence of spectral selectors, providing a new tool for studying Legendrian submanifolds and contactomorphism spaces.

Findings

Orderability is equivalent to spectral selectors existence.

Existence of Reeb chords between orderable Legendrians.

Applications to longstanding conjectures and contact topology metrics.

Abstract

We study the notion of orderability of isotopy classes of Legendrian submanifolds and their universal covers, with some weaker results concerning spaces of contactomorphisms. Our main result is that orderability is equivalent to the existence of spectral selectors analogous to the spectral invariants coming from Lagrangian Floer Homology. A direct application is the existence of Reeb chords between any closed Legendrian submanifolds of a same orderable isotopy class. Other applications concern the Sandon conjecture, the Arnold chord conjecture, Legendrian interlinking, the existence of time-functions and the study of metrics due to Hofer-Chekanov-Shelukhin, Colin-Sandon, Fraser-Polterovich-Rosen and Nakamura.