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Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

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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 (Acta. Math. 141(1–2):17–34, 1978). 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.

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Acknowledgments

I would like to thank my Ph.D. adviser Gabriel P. Paternain for many helpful discussions. I am also extremely grateful to Alberto Abbondandolo, Peter Albers and Urs Frauenfelder, together with all the participants of the 2009–2010 Cambridge seminar on Rabinowitz Floer homology, for several stimulating remarks and insightful suggestions, and for pointing out errors in previous drafts of this work. This work forms part of my PhD thesis [38]. Finally, I am grateful to Irida Altman and the anonymous referees for their useful comments on making the paper more readable.

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  1. Department of Mathematics, ETH Zürich, Switzerland

    Will J. Merry

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Merry, W.J. Lagrangian Rabinowitz Floer homology and twisted cotangent bundles. Geom Dedicata 171, 345–386 (2014). https://doi.org/10.1007/s10711-013-9903-9

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