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Moscow Mathematical Journal, 2003, Volume 3, Number 2, Pages 593–619
DOI: https://doi.org/10.17323/1609-4514-2003-3-2-593-619 (Mi mmj101) 		 
This article is cited in 40 scientific papers (total in 40 papers)

Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory

G. P. Paternaina, L. V. Polterovichb, K. Siburgc

a Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge
b Tel Aviv University, School of Mathematical Sciences
c Ruhr-Universität Bochum
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DOI: https://doi.org/10.17323/1609-4514-2003-3-2-593-619
Abstract: The paper establishes a link between symplectic topology and Aubry–Mather theory. We show that certain Lagrangian submanifolds lying in an optical hypersurface cannot be deformed into the domain bounded by the hypersurface. Even when this rigidity fails, we find that the intersection between the deformed Lagrangian submanifold and the hypersurface always contains a dynamically significant set related to Aubry–Mather theory. This phenomenon, although in a weaker form, still persists in the non-optical case.
Key words and phrases: Lagrangian submanifold, optical hypersurface, characteristic foliation, Liouville class, symplectic shape, generating function, Aubry set.
Received: July 11, 2002
Bibliographic databases:
MSC: 53D12, 37J50, 57R17, 57R30
Language: English
Citation: G. P. Paternain, L. V. Polterovich, K. Siburg, “Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory”, Mosc. Math. J., 3:2 (2003), 593–619
Citation in format AMSBIB
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\paper Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry--Mather theory
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\vol 3
\issue 2
\pages 593--619
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