Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

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 [Mos78]. 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.

The Omega limit set of a family of chords

“…For periodic orbits this functional was first considered in [9], Equation 2.7, and a Floer theory for this functional was developed in [5]. Extensions to the chord case can be found in [7,8]. Some of the arguments that will be used here were earlier explored in [1].…”

Existence of either a periodic collisional orbit or infinitely many consecutive collision orbits in the planar circular restricted three-body problem

“…The grading is obtained by the transversal Maslov index at a chord. One defines a boundary operator As the notation suggests the resulting homology is independent of the choice of the Hamiltonian H as well as on the choice of the family of ω-compatible almost complex structures needed to define the gradient [4,10]. The superscript + is added, because we suppose that τ only takes positive values.…”

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

“…is well defined (see [Mer10]). Since {ϕ t } is a loop the number of critical points of the underlying Rabinowitz action functional grows linearly with the action value.…”

“…According to [Mer10,Theorem B] the Rabinowitz Floer homology in positive degrees of the path {ψ t } is isomorphic to the homology of the based loop space. It follows from Gromov's theorem [Gro78,Gro07] (see also [Pat99]) that if the homology of the loop space grows at most linearly in action then it also grows at most linearly in degree.…”