Skip to main content
SpringerLink
Log in

Some remarks on Lagrangian tori

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

We illustrate the power of Floer theory for Lagrangian submanifolds through some of its applications in symplectic topology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abouzaid, K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Quantum cohomology and split generation in Lagrangian Floer theory. In preparation.

  2. Albers P.: On the extrinsic topology of Lagrangian submanifolds. Int. Math. Res. Not. IMRN 38, 2341–2371 (2005)

    Article  MathSciNet  Google Scholar 

  3. P. Biran and O. Cornea, A Lagrangian quantum homology. In: New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes 49, Amer. Math. Soc., Providence, RI, 2009, 1–44.

  4. Biran P., Cornea O.: Rigidity and uniruling for Lagrangian submanifolds. Geom. Topol. 13, 2881–2989 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgeois F., Eliashberg Y., Hofer H., Wysocki K., Zehnder E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Calabi, On the group of automorphisms of a symplectic manifold. In: Problem in Analysis (Lectures at the Sympos. in Honor of Salomon Bochner, Princeton University, Princeton, NJ, 1969), Princeton University Press, Princeton, NJ, 1970, 1–26.

  7. Chekanov Yu.: Lagrangian intersections, symplectic energy and areas of holomorphic curves. Duke Math. J. 95, 213–226 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Entov M., Polterovich L.: Calabi quasimorphisms and quantum homology. Int. Math. Res. Not. IMRN 2003, 1635–1676 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Entov M., Polterovich L.: Quasi-states and symplectic intersections. Comment. Math. Helv. 81, 75–99 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Entov M., Polterovich L.: Rigid subsets of symplectic manifolds. Compos. Math. 145, 773–826 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology. In: Toric Topology, Contemp. Math. 460, Amer. Math. Soc., Providence, RI, 2008, 47–70.

  12. Floer A.: Morse theory for Lagrangian intersections. J. Differential Geom. 28, 513–547 (1988)

    MATH  MathSciNet  Google Scholar 

  13. Floer A., Hofer H., Wysocki K.: Applications of symplectic homology. I. Math. Z. 217, 577–606 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory: Anomaly and obstruction. Part I. AMS/IP Studies in Advaneced Math. 46, Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2009.

  15. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory: Anomaly and obstruction. Part II. AMS/IP Studies in Advaneced Math. 46, Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2009.

  16. Fukaya K., Oh Y.-G., Ohta H., Ono K.: Lagrangian Floer theory on compact toric manifolds. I. Duke Math. J. 151, 23–174 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds. II: Bulk deformations. Selecta Math. (N.S.) 17 (2011), 609–711.

  18. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory and mirror symmetry on compact toric manifolds. arXiv:1009.1648 [math.SG], 2010.

  19. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds: Survey. In: Surveys in Differential Geometry, Vol. XVII, Surv. Differ. Geom. 17, International Press, Boston, MA, 2012, 229–298.

  20. Fukaya K., Oh Y.-G., Ohta H., Ono K.: Displacement of polydisks and Lagrangian Floer theory. J. Symplectic Geom. 11, 231–268 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory. arXiv:1105.5123 [math.SG], 2011.

  22. Hofer H.: On the topological properties of symplectic maps. Proc. Royal Soc. Edinburgh Sect. A 115, 25–38 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hu S., Lalonde F., Leclercq R.: Homological Lagrangian monodromy. Geom. Topol. 15, 1617–1650 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lalonde F., McDuff D.: The geometry of symplectic energy. Ann. of Math. (2) 141, 349–371 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Oh Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Comm. Pure Appl. Math. 46, 949–993 (1993)

    Article  MATH  Google Scholar 

  26. Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. II. (\({CP^n}\) , \({RP^n}\)). Comm. Pure Appl. Math. 46 (1993), 995–1012.

  27. Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. III. Arnol'd-Givental conjecture. In: The Floer Memorial Volume, Progr. Math. 133, Birkhäuser, Basel, 1995, 555–573.

  28. Polterovich L.: Symplectic displacement energy for Lagrangian submanifolds. Ergodic Theory Dynam. Systems 13, 357–367 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yau M.-L.: Monodromy and isotopy of monotone Lagrangian tori. Math. Res. Lett. 16, 531–541 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kaoru Ono

Authors
  1. Kaoru Ono
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kaoru Ono.

Additional information

To Professor Andrzej Granas

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ono, K. Some remarks on Lagrangian tori. J. Fixed Point Theory Appl. 17, 221–237 (2015). https://doi.org/10.1007/s11784-015-0248-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-015-0248-x

Mathematics Subject Classification

Keywords

Search

Navigation