Invariant distances on spaces of Legendrians

TL;DR

This paper introduces new unbounded invariant distances on Legendrian isotopy classes, demonstrating their unboundedness without positive loops and establishing their discreteness.

Contribution

It constructs the first unbounded invariant distances on Legendrian isotopy classes without positive loops and proves their discreteness.

Findings

Unbounded invariant distances are constructed on certain Legendrian classes.

Such distances can be unbounded without positive loops of contactomorphisms.

Invariant distances on Legendrian classes are necessarily discrete.

Abstract

We construct new unbounded invariant distances on the universal cover of certain Legendrian isotopy classes. This is the first instance where unboundedness of an invariant distance is obtained without assuming the existence of a positive loop of contactomorphisms. We also show that invariant distances on Legendrian isotopy classes have to be discrete.