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Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory

Published online by Cambridge University Press:  31 August 2017

LINO AMORIM ,
YONG–GEUN OH  and
JOANA OLIVEIRA DOS SANTOS
LINO AMORIM
Affiliation:
Mathematics Department, 138 Cardwell Hall, 1228 N. 17th Street, Manhattan, KS 66506-2602, U.S.A. e-mail: lamorim@ksu.edu
YONG–GEUN OH
Affiliation:
Center for Geometry and Physics, Institute for Basic Sciences (IBS), & Department of Mathematics, POSTECH, Pohang 37673, Korea. e-mail: yongoh1@postech.ac.kr
JOANA OLIVEIRA DOS SANTOS
Affiliation:
Mathematics Department, 138 Cardwell Hall, 1228 N. 17th Street, Manhattan, KS 66506-2602, U.S.A. e-mail: jamorim@ksu.edu
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Abstract

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 411 - 434
Copyright
Copyright © Cambridge Philosophical Society 2017 

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