Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

TL;DR

This paper develops Lagrangian Rabinowitz Floer homology to study rigidity problems in symplectic geometry, demonstrating the existence of infinitely many relative leaf-wise intersection points in certain twisted cotangent bundles.

Contribution

It introduces a new Floer homology framework for Liouville domains and extends it to the virtually contact setting, enabling computations in twisted cotangent bundles.

Findings

Computed Lagrangian Rabinowitz Floer homology for specific level sets of Tonelli Hamiltonians.

Proved the existence of infinitely many relative leaf-wise intersection points under generic conditions.

Linked the homology to the Arnold Chord Conjecture and symplectic rigidity phenomena.

Abstract

We study the following rigidity problem in symplectic geometry:can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser. Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the `virtually contact' setting. By means of an Abbondandolo-Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.