The persistence of a relative Rabinowitz-Floer complex

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.

Version 4 is substantially longer than prior versions because many details have been added (including new appendices A and B), as well as multiple explanatory figures. The conditions for Theorem 1.8 on interlinkedness have been corrected

References in corpus (4)

• Holomorphic curves for Legendrian surgery
• Local rigidity, contact homeomorphisms, and conformal factors
• Augmentations and immersed Lagrangian fillings
• Geometry and analysis of contact instantons and entanglement of Legendrian links I

Cited by in corpus (1)

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