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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
Affiliation:
Mathematics Department, 138 Cardwell Hall, 1228 N. 17th Street, Manhattan, KS 66506-2602, U.S.A. e-mail: [email protected]
YONG–GEUN OH
Affiliation:
Center for Geometry and Physics, Institute for Basic Sciences (IBS), & Department of Mathematics, POSTECH, Pohang 37673, Korea. e-mail: [email protected]
JOANA OLIVEIRA DOS SANTOS
Affiliation:
Mathematics Department, 138 Cardwell Hall, 1228 N. 17th Street, Manhattan, KS 66506-2602, U.S.A. e-mail: [email protected]

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
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[Ab1] Abouzaid, M. A cotangent fibre generates the Fukaya category. Adv. Math. 228, no. 2 (2011), 894939.Google Scholar
[Ab2] Abouzaid, M. Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189, no. 2 (2012), 251313.Google Scholar
[Ab3] Abouzaid, M. On the Wrapped Fukaya category and based loops. J. Symplectic Geom. 10, no. 1 (2012), 2779.Google Scholar
[AS] Abouzaid, M. and Seidel, P. An open string analogue of Viterbo functoriality. Geometry and Topology 14 (2010), 627718.Google Scholar
[Arn] Arnaud, M.–C. On a theorem due to Birkhoff. Geom. Funct. Anal. 120 (2010), 13071316.Google Scholar
[Be1] Bernard, P. Existence of C 1,1 critical sub-solutions of the Hamilton–Jacobi equations on compact manifolds. Ann. Sci. École Norm. Sup. 40, no. 3 (2007), 445452.Google Scholar
[Be2] Bernard, P. Symplectic Aspects of Mather theory. Duke Math. J. 136, no. 3 (2007), 401420.Google Scholar
[BO1] Bernard, P. and Oliveira dos Santos, J. A geometric definition of the Aubry–Mather set. J. Top. Anal. 2, no. 3 (2010), 385393.Google Scholar
[BO2] Bernard, P. and Oliveira dos Santos, J. A geometric definition of the Mañé-Mather set and a Theorem of Marie–Claude Arnaud. Math. Proc. Camb. Phil. Soc. 152 (2012), 167178.Google Scholar
[BS] Buhovsky, L. and Seyfaddini, S. Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symp. Geom. 11, no. 1 (2013), 3752.Google Scholar
[C] Chaperon, M. Lois de conservation et géométrie symplectique. Comptes rendus de l'Académie des sciences. Série 1, Mathématique 312, no. 4 (1991), 345348.Google Scholar
[dR] de Rham, G. Differentiable Manifolds. A Series of Comp. Studies in Math. 266 (Springer Verlag, Berlin-Heidelberg-New York, 1984).Google Scholar
[El] Eliashberg, Y. A theorem on the structure of wave fronts and its application in symplectic topology (in Russian). Funkstsional. Anal. i Prilozhen. 21, no. 3 (1987), 6572.Google Scholar
[EG] Evans, L. and Gariepy, R. Measure theory and fine properties of functions. Stud. Adv. Math. (CRC Press, New York, 1992).Google Scholar
[Fe] Federer, H. Geometric Measure Theory, Classics in Math. Springer-Verlag, Berlin-Heidelberg-New York, 1969.Google Scholar
[FOOO] Fukaya, K., Oh, Y.–G., Ohta, H. and Ono, K. Lagrangian intersection Floer theory-anomaly and obstruction I - II. Stud. Adv. Math., vol. 46 (Amer. Math. Soc., International Press, 2009).Google Scholar
[FSS] Fukaya, K., Seidel, P. and Smith, I. Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172 (2008), 127.Google Scholar
[Hor] Hörmander, L. Fourier integral operators I. Acta Math. 127 (1971), 79183.Google Scholar
[HLS] Humiliére, V., Leclerq, R. and Seyfaddini, S. Coisotropic rigidity and C 0-symplectic geometry. Duke Math. J. 164, no. 4 (2015), 767799.Google Scholar
[KO] Kasturirangan, R. and Oh, Y.–G. Floer homology of open subsets and a relative version of Arnold's conjecture. Math. Z. 236, no. 1 (2001), 151189.Google Scholar
[Kra] Kragh, T. Parametrized ring-spectra and the nearby Lagrangian conjecture. Geometry and Topology 17, no. 2 (2013), 639731.Google Scholar
[LauS] Laudenbach, F. and Sikorav, J.–C. Persistence of intersection with the zero section during a Hamiltonian isotopy into a cotangent bundle. Invent. Math 82, no. 2 (1985), 349357.Google Scholar
[Mu] Müller, S. The group of Hamiltonian homeomorphisms in the L -norm. J. Korean Math. Soc. 45, no. 6 (2008), 17691784.Google Scholar
[N] Nadler, D. Microlocal branes are constructible sheaves. Selecta Math. 15, no. 4 (2009), 563619.Google Scholar
[Oh1] Oh, Y.–G. Symplectic topology as the geometry of action functional, I. J. Differential Geom. 46 (1997) 499577.Google Scholar
[Oh2] Oh, Y.–G. Symplectic topology as the geometry of action functional, II. Commun. Anal. Geom. 7 (1999), 155.Google Scholar
[Oh3] Oh, Y.–G. Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds in “The Breadth of Symplectic and Poisson Geometry”. Prog. Math. 232 (Birkhäuser, Boston, 2005), 525570.Google Scholar
[Oh4] Oh, Y.–G. Locality of continuous Hamiltonian flows and Lagrangian intersection with the conormal of open subsets. J. Gökova Geom. Top. 1 (2007), 132.Google Scholar
[Oh5] Oh, Y.–G. Floer mini-max theory, the Cerf diagram, and the spectral invariants. J. Korean Mah. Soc. 46 (2009), 363447.Google Scholar
[Oh6] Oh, Y.–G. Symplectic topology and Floer homology I & II. New Mathematical Monographs, no. 28 and 29 (Cambridge University Press, Cambridge, 2015).Google Scholar
[OM] Oh, Y.–G. and Müller, S. The group of Hamiltonian homeomorphisms and C 0 symplectic topology. J. Symp. Geom. 5 (2007), 167219.Google Scholar
[PPS] Paternain, G., Polterovich, L. and Siburg, K. Boundary rigidity for Lagrangian submanifolds, non-removable intersections and Aubry–Mather theory. Mosc. Math. J. 3, no. 2 (2003), 593619.Google Scholar
[Se] Seidel, P. Fukaya categories and Picard–Lefschetz theory. Zürich Lec. Advanced Math. (European Math. Soc., Zürich, 2008).Google Scholar
[Sik] Sikorav, J. C. Problémes d'intersections et de points fixes en géométrie hamiltonienne. Comment. Math. Helv. 62 (1987), 6273.Google Scholar
[V1] Viterbo, C. Symplectic topology as the geometry of generating functions. Math. Ann. 292 (1992), 685710.Google Scholar
[V2] Viterbo, C. On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonian flows. Internat. Math. Res. Notices, (2006), article ID 34028. Erratum, ibid, (2006), article ID 38784.Google Scholar