Symplectic embeddings and continued fractions: a survey



As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

Keywords and phrases:

symplectic embedding symplectic capacity continued fractions symplectic ellipsoid symplectic packing lattice points in triangles 

Mathematics Subject Classification (2000):

53D05 32S25 11J70 


  1. 1.
    P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal., 7 (1997), 420–437.Google Scholar
  2. 2.
    P. Biran, A stability property of symplectic packing, Invent. Math., 136 (1999), 123–155.Google Scholar
  3. 3.
    P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, arXiv:0708.4221.Google Scholar
  4. 4.
    K. Cieliebak, A. Floer and H. Hofer, Symplectic homology. II. A general construction, Math. Z., 218 (1995), 103–122.Google Scholar
  5. 5.
    K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, In: Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, 2007, pp. 1–44; arXiv:math/0506191.Google Scholar
  6. 6.
    A. Craw and M. Reid, How to calculate A-Hilb \({\mathbb{C}}^3\), arXiv:math/9909085.Google Scholar
  7. 7.
    I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355–378.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Y. Eliashberg, Rigidity of symplectic and contact structures, Abstracts of reports to the 7th Leningrad International Topology Conference, 1982.Google Scholar
  9. 9.
    A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology. I, Math. Z., 217 (1994), 577–606.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton Univ. Press, 1993.Google Scholar
  11. 11.
    M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent.Math., 82 (1985), 307–347.Google Scholar
  12. 12.
    L. Guth, Symplectic embeddings of polydisks, arXiv:0709.1957.Google Scholar
  13. 13.
    G.H. Hardy and J.E. Littlewood, Some problems of Diophantine approximation: The lattice-points of a right-angled triangle, Proc. London Math. Soc., s2-20 (1922), 15–36.Google Scholar
  14. 14.
    G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press, 1938.Google Scholar
  15. 15.
    R. Hind and E. Kerman, New obstructions to symplectic embeddings, arXiv:0906.4296.Google Scholar
  16. 16.
    H. Hofer, Estimates for the energy of a symplectic map, Comment.Math. Helv., 68 (1993), 48–72.Google Scholar
  17. 17.
    H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, In: Analysis, et cetera, (eds. P.H. Rabinowitz and E. Zehnder), Academic Press, Boston, MA, 1990, pp. 405–429.Google Scholar
  18. 18.
    M. Hutchings and C. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. I, arXiv:math/0701300, to appear in J. Symplectic Geom.Google Scholar
  19. 19.
    F. Lalonde and D. McDuff, The geometry of symplectic energy, Ann. of Math. (2), 141 (1995), 349–371.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    B.-H. Li and T.-J. Li, Symplectic genus, minimal genus and diffeomorphisms, Asian J. Math., 6 (2002), 123–144.MATHMathSciNetGoogle Scholar
  21. 21.
    D. McDuff, Symplectic embeddings of 4-dimensional ellipsoids, to appear in J. Topol. (2009).Google Scholar
  22. 22.
    D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math., 115 (1994), 405–429.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. McDuff and F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids, in preparation.Google Scholar
  24. 24.
    E. Opshtein, Maximal symplectic packings of \({\mathbb{P}}^{2}\), arXiv:math/0610677.Google Scholar
  25. 25.
    P. Popescu-Pampu, The geometry of continued fractions and the topology of surface singularities, arXiv:math/0506432.Google Scholar
  26. 26.
    F. Schlenk, Embedding Problems in Symplectic Geometry, de Gruyter Exp. Math., de Gruyter, Berlin, 2005.Google Scholar
  27. 27.
    L. Traynor, Symplectic packing constructions, J. Differential Geom., 42 (1995), 411–429.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2009

Authors and Affiliations

  1. 1.Department of MathematicsBarnard College, Columbia UniversityNew YorkUSA

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