$C^0$-limits of Legendrians and positive loops

TL;DR

This paper proves that Legendrian submanifolds are preserved under $C^0$-limits of contactomorphisms if the image remains smooth, and shows non-Legendrian submanifolds admit positive loops, advancing understanding of Legendrian topology.

Contribution

It establishes the Legendrian property is preserved under $C^0$-limits of contactomorphisms and introduces positive loops for non-Legendrian submanifolds, with a parametric refinement of existing degeneracy results.

Findings

Legendrian submanifolds are preserved under $C^0$-limits of contactomorphisms.

Non-Legendrian submanifolds admit positive loops.

Refinement of Rosen--Zhang result on pseudo-norm degeneracy.

Abstract

We show that the image of a Legendrian submanifold under a homeomorphism that is the $C^0$-limit of a sequence of contactomorphisms is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen--Zhang result on the degeneracy of the Chekanov--Hofer--Shelukhin pseudo-norm for non-Legendrians.