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

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.

V2: Added two references and Remark 2.7. V3: Removed one lemma in Section 3.2, resulting in slightly adjusted subsequent lemmas from that section. V4: Weakened Theorem C to closed Legendrians

References in corpus (6)

• Non-negative Legendrian isotopy in $S{T}^{\ast }M$
• Legendrian links, causality, and the Low conjecture
• Chekanov's dichotomy in contact topology
• Observations on the Hofer distance between closed subsets
• Local rigidity, contact homeomorphisms, and conformal factors
• The persistence of a relative Rabinowitz-Floer complex