Skip to main content

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


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.

Hide All
[Ab1] Abouzaid M. A cotangent fibre generates the Fukaya category. Adv. Math. 228, no. 2 (2011), 894939.
[Ab2] Abouzaid M. Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189, no. 2 (2012), 251313.
[Ab3] Abouzaid M. On the Wrapped Fukaya category and based loops. J. Symplectic Geom. 10, no. 1 (2012), 2779.
[AS] Abouzaid M. and Seidel P. An open string analogue of Viterbo functoriality. Geometry and Topology 14 (2010), 627718.
[Arn] Arnaud M.–C. On a theorem due to Birkhoff. Geom. Funct. Anal. 120 (2010), 13071316.
[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.
[Be2] Bernard P. Symplectic Aspects of Mather theory. Duke Math. J. 136, no. 3 (2007), 401420.
[BO1] Bernard P. and Oliveira dos Santos J. A geometric definition of the Aubry–Mather set. J. Top. Anal. 2, no. 3 (2010), 385393.
[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.
[BS] Buhovsky L. and Seyfaddini S. Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symp. Geom. 11, no. 1 (2013), 3752.
[C] Chaperon M. Lois de conservation et géométrie symplectique. Comptes rendus de l'Académie des sciences. Série 1, Mathématique 312, no. 4 (1991), 345348.
[dR] de Rham G. Differentiable Manifolds. A Series of Comp. Studies in Math. 266 (Springer Verlag, Berlin-Heidelberg-New York, 1984).
[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.
[EG] Evans L. and Gariepy R. Measure theory and fine properties of functions. Stud. Adv. Math. (CRC Press, New York, 1992).
[Fe] Federer H. Geometric Measure Theory, Classics in Math. Springer-Verlag, Berlin-Heidelberg-New York, 1969.
[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).
[FSS] Fukaya K., Seidel P. and Smith I. Exact Lagrangian submanifolds in simply-connected cotangent bundles. Invent. Math. 172 (2008), 127.
[Hor] Hörmander L. Fourier integral operators I. Acta Math. 127 (1971), 79183.
[HLS] Humiliére V., Leclerq R. and Seyfaddini S. Coisotropic rigidity and C 0-symplectic geometry. Duke Math. J. 164, no. 4 (2015), 767799.
[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.
[Kra] Kragh T. Parametrized ring-spectra and the nearby Lagrangian conjecture. Geometry and Topology 17, no. 2 (2013), 639731.
[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.
[Mu] Müller S. The group of Hamiltonian homeomorphisms in the L -norm. J. Korean Math. Soc. 45, no. 6 (2008), 17691784.
[N] Nadler D. Microlocal branes are constructible sheaves. Selecta Math. 15, no. 4 (2009), 563619.
[Oh1] Oh Y.–G. Symplectic topology as the geometry of action functional, I. J. Differential Geom. 46 (1997) 499577.
[Oh2] Oh Y.–G. Symplectic topology as the geometry of action functional, II. Commun. Anal. Geom. 7 (1999), 155.
[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.
[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.
[Oh5] Oh Y.–G. Floer mini-max theory, the Cerf diagram, and the spectral invariants. J. Korean Mah. Soc. 46 (2009), 363447.
[Oh6] Oh Y.–G. Symplectic topology and Floer homology I & II. New Mathematical Monographs, no. 28 and 29 (Cambridge University Press, Cambridge, 2015).
[OM] Oh Y.–G. and Müller S. The group of Hamiltonian homeomorphisms and C 0 symplectic topology. J. Symp. Geom. 5 (2007), 167219.
[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.
[Se] Seidel P. Fukaya categories and Picard–Lefschetz theory. Zürich Lec. Advanced Math. (European Math. Soc., Zürich, 2008).
[Sik] Sikorav J. C. Problémes d'intersections et de points fixes en géométrie hamiltonienne. Comment. Math. Helv. 62 (1987), 6273.
[V1] Viterbo C. Symplectic topology as the geometry of generating functions. Math. Ann. 292 (1992), 685710.
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 4
Total number of PDF views: 26 *
Loading metrics...

Abstract views

Total abstract views: 182 *
Loading metrics...

* Views captured on Cambridge Core between 31st August 2017 - 19th February 2018. This data will be updated every 24 hours.