The persistence of a relative Rabinowitz-Floer complex

TL;DR

This paper refines the invariance properties of Legendrian contact homology, constructs a relative Rabinowitz-Floer complex with persistent homology analysis, and applies these tools to prove bounds and non-displaceability results for Legendrian submanifolds.

Contribution

It introduces a quantitative invariance of the Legendrian contact homology algebra and constructs a relative Rabinowitz-Floer complex with persistent homology, advancing Legendrian invariants.

Findings

Displacement energy bounds for Legendrian submanifolds.

Proof of Rosen and Zhang's non-degeneracy conjecture.

Non-displaceability of Legendrian real-projective space.

Abstract

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame isomorphism which is similar to a bifurcation invariance-proof for a contactization contact manifold. We use this result to construct a relative version of the Rabinowitz-Floer complex defined for Legendrians that also satisfies a quantitative invariance, and study its persistent homology barcodes. We apply these barcodes to prove several results, including: displacement energy bounds for Legendrian submanifolds in terms of the oscillatory norms of the contact Hamiltonians; a proof of Rosen and Zhang's non-degeneracy conjecture for the Shelukhin--Chekanov--Hofer metric on Legendrian submanifolds; and, the non-displaceability of the standard Legendrian real-projective space inside the contact real-projective space.