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We study the following rigidity problem in symplectic geometry:can onedisplace a Lagrangian submanifold from a hypersurface? We relate this to theArnold Chord Conjecture, and introduce a refined question about the existenceof relative leaf-wise intersection points, which are the Lagrangian-theoreticanalogue of the notion of leaf-wise intersection points defined by Moser. Ourtool is Lagrangian Rabinowitz Floer homology, which we define first forLiouville domains and exact Lagrangian submanifolds with Legendrian boundary.We then extend this to the `virtually contact' setting. By means of anAbbondandolo-Schwarz short exact sequence we compute the Lagrangian RabinowitzFloer homology of certain regular level sets of Tonelli Hamiltonians ofsufficiently high energy in twisted cotangent bundles, where the Lagrangiansare conormal bundles. We deduce that in this situation a generic Hamiltoniandiffeomorphism has infinitely many relative leaf-wise intersection points.

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles