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

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 generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

v4: final version; to appear in Math. Proc. Camb. Phil. Soc

References in corpus (4)

• On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows
• Exact Lagrangian submanifolds in simply-connected cotangent bundles
• Coisotropic rigidity and C^0-symplectic geometry
• Locality of continuous Hamiltonian flows and Lagrangian intersections with the conormal of open subsets