The embedding capacity of $4$-dimensional symplectic ellipsoids

Abstract

This paper calculates the function $c(a)$ whose value at $a$ is the infimum of the size of a ball that contains a symplectic image of the ellipsoid $E(1,a)$. (Here $a \ge 1$ is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of $c(a)$ is surprisingly rich. The volume constraint implies that $c(a)$ is always greater than or equal to the square root of $a$, and it is not hard to see that this is equality for large $a$. However, for $a$ less than the fourth power $\tau^4$ of the golden ratio, $c(a)$ is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval $[ \tau^4,7 ]$ we find $c(a) = (a+1)/3$. For $a \ge 7$, the function $c(a)$ coincides with the square root except on a finite number of intervals where it is again piecewise linear.
The embedding constraints coming from embedded contact homology give rise to another capacity function $c_{\rm ECH}$ which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that % $c_{\rm ECH}(a) \le c(a)$ for all $a$. We show here that $c_{\rm ECH}(a) \ge c(a)$ for all $a$.

  • [B] Go to document P. Biran, "Symplectic packing in dimension $4$," Geom. Funct. Anal., vol. 7, iss. 3, pp. 420-437, 1997.
    @article {B, MRKEY = {1466333},
      AUTHOR = {Biran, Paul},
      TITLE = {Symplectic packing in dimension {$4$}},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {7},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {420--437},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {57R40 (57R15 58F05)},
      MRNUMBER = {1466333},
      MRREVIEWER = {Hansj{ö}rg Geiges},
      DOI = {10.1007/s000390050014},
      ZBLNUMBER = {0892.53022},
      }
  • [B-99] Go to document P. Biran, "Constructing new ample divisors out of old ones," Duke Math. J., vol. 98, iss. 1, pp. 113-135, 1999.
    @article {B-99, MRKEY = {1687571},
      AUTHOR = {Biran, Paul},
      TITLE = {Constructing new ample divisors out of old ones},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {98},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {113--135},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {14J26 (14C20 14E07)},
      MRNUMBER = {1687571},
      MRREVIEWER = {Takao Fujita},
      DOI = {10.1215/S0012-7094-99-09803-4},
      ZBLNUMBER = {0961.14005},
      }
  • [B-01] P. Biran, "From symplectic packing to algebraic geometry and back," in European Congress of Mathematics, Vol. II, Basel: Birkhäuser, 2001, vol. 202, pp. 507-524.
    @incollection {B-01, MRKEY = {1909952},
      AUTHOR = {Biran, Paul},
      TITLE = {From symplectic packing to algebraic geometry and back},
      BOOKTITLE = {European {C}ongress of {M}athematics, {V}ol. {II}},
      VENUE={{B}arcelona, 2000},
      SERIES = {Progr. Math.},
      VOLUME = {202},
      PAGES = {507--524},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2001},
      MRCLASS = {53D35 (14J25 57R17 57R40)},
      MRNUMBER = {1909952},
      MRREVIEWER = {Hansj{ö}rg Geiges},
      ZBLNUMBER = {1047.53054},
      }
  • [BH] Go to document O. Buse and R. Hind, "Symplectic embeddings of ellipsoids in dimension greater than four," Geom. and Top., vol. 15, pp. 2091-2110, 2011.
    @article{BH,
      author={Buse, O. and Hind, R.},
      TITLE={Symplectic embeddings of ellipsoids in dimension greater than four},
      JOURNAL={Geom. and Top.},
      VOLUME={15},
      PAGES={2091--2110},
      YEAR={2011},
      ZBLNUMBER = {05996759},
      DOI = {10.2140/gt.2011.15.2091},
     }
  • [CHLS] Go to document K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk, "Quantitative Symplectic Geometry," in Dynamics, Ergodic Theory, and Geometry, Cambridge: Cambridge Univ. Press, 2007, vol. 54, pp. 1-44.
    @incollection {CHLS, MRKEY = {2369441},
      AUTHOR = {Cieliebak, Kai and Hofer, Helmut and Latschev, Janko and Schlenk, Felix},
      TITLE = {Quantitative Symplectic Geometry},
      BOOKTITLE = {Dynamics, Ergodic Theory, and Geometry},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {54},
      PAGES = {1--44},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2007},
      MRCLASS = {53D35 (37J05 57R17)},
      MRNUMBER = {2369441},
      MRREVIEWER = {Michael J. Usher},
      DOI = {10.1017/CBO9780511755187.002},
      ZBLNUMBER = {1143.53341},
      }
  • [EH] Go to document I. Ekeland and H. Hofer, "Symplectic topology and Hamiltonian dynamics. II," Math. Z., vol. 203, iss. 4, pp. 553-567, 1990.
    @article {EH, MRKEY = {1044064},
      AUTHOR = {Ekeland, Ivar and Hofer, Helmut},
      TITLE = {Symplectic topology and {H}amiltonian dynamics. {II}},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {203},
      YEAR = {1990},
      NUMBER = {4},
      PAGES = {553--567},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {58F05 (53C15 57R15 58E05)},
      MRNUMBER = {1044064},
      MRREVIEWER = {Michel Willem},
      DOI = {10.1007/BF02570756},
      ZBLNUMBER = {0729.53039},
      }
  • [Gu] Go to document L. Guth, "Symplectic embeddings of polydisks," Invent. Math., vol. 172, iss. 3, pp. 477-489, 2008.
    @article {Gu, MRKEY = {2393077},
      AUTHOR = {Guth, Larry},
      TITLE = {Symplectic embeddings of polydisks},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {172},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {477--489},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {53D35 (57R17)},
      MRNUMBER = {2393077},
      MRREVIEWER = {Janko Latschev},
      DOI = {10.1007/s00222-007-0103-9},
      ZBLNUMBER = {1153.53060},
      }
  • [HW] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Third ed., Oxford, at the Clarendon Press, 1954.
    @book {HW, MRKEY = {0067125},
      AUTHOR = {Hardy, G. H. and Wright, E. M.},
      TITLE = {An Introduction to the Theory of Numbers},
      EDITION = {Third},
      PUBLISHER = {Oxford, at the Clarendon Press},
      YEAR = {1954},
      PAGES = {xvi+419},
      MRCLASS = {10.0X},
      MRNUMBER = {0067125},
      MRREVIEWER = {N. G. de Bruijn},
      ZBLNUMBER = {0058.03301},
      }
  • [HK] R. Hind and E. Kerman, New obstructions to symplectic embeddings.
    @misc{HK,
      author={Hind, R. and Kerman, E.},
      TITLE={New obstructions to symplectic embeddings},
      ARXIV={0906.4296},
     }
  • [Hu1] Go to document M. Hutchings, "Quantitative embedded contact homology," J. Differential Geom., vol. 88, iss. 2, pp. 231-266, 2011.
    @article{Hu1,
      author = {Hutchings, Michael},
      TITLE = {Quantitative embedded contact homology},
      JOURNAL = {J. Differential Geom.},
      VOLUME = {88},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {231--266},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53D42},
      MRNUMBER = {2838266},
      ZBLNUMBER = {06011499},
      URL ={http://projecteuclid.org/getRecord?id=euclid.jdg/1320067647},
      }
  • [Hu2] Go to document M. Hutchings, "Recent progress on symplectic embedding problems in four dimensions," Proc. Natl. Acad. Sci. USA, vol. 108, iss. 20, pp. 8093-8099, 2011.
    @article {Hu2, MRKEY = {2806644},
      AUTHOR = {Hutchings, Michael},
      TITLE = {Recent progress on symplectic embedding problems in four dimensions},
      JOURNAL = {Proc. Natl. Acad. Sci. USA},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {108},
      YEAR = {2011},
      NUMBER = {20},
      PAGES = {8093--8099},
      ISSN = {1091-6490},
      CODEN = {PNASFB},
      MRCLASS = {53Dxx},
      MRNUMBER = {2806644},
      DOI = {10.1073/pnas.1018622108},
      }
  • [HT] Go to document M. Hutchings and C. H. Taubes, "Gluing pseudoholomorphic curves along branched covered cylinders. I," J. Symplectic Geom., vol. 5, iss. 1, pp. 43-137, 2007.
    @article {HT, MRKEY = {2371184},
      AUTHOR = {Hutchings, Michael and Taubes, Clifford Henry},
      TITLE = {Gluing pseudoholomorphic curves along branched covered cylinders. {I}},
      JOURNAL = {J. Symplectic Geom.},
      FJOURNAL = {The Journal of Symplectic Geometry},
      VOLUME = {5},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {43--137},
      ISSN = {1527-5256},
      MRCLASS = {53D40 (53D45 57R17 57R57)},
      MRNUMBER = {2371184},
      MRREVIEWER = {Michael J. Usher},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jsg/1197491304},
      ZBLNUMBER = {1157.53047},
      }
  • [LL2] B. Li and T. Li, "Symplectic genus, minimal genus and diffeomorphisms," Asian J. Math., vol. 6, iss. 1, pp. 123-144, 2002.
    @article {LL2, MRKEY = {1902650},
      AUTHOR = {Li, Bang-He and Li, Tian-Jun},
      TITLE = {Symplectic genus, minimal genus and diffeomorphisms},
      JOURNAL = {Asian J. Math.},
      FJOURNAL = {The Asian Journal of Mathematics},
      VOLUME = {6},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {123--144},
      ISSN = {1093-6106},
      MRCLASS = {57R17 (57R57 57R95)},
      MRNUMBER = {1902650},
      MRREVIEWER = {Ivan Smith},
      ZBLNUMBER = {1008.57024},
      }
  • [LL] Go to document T. Li and A. Liu, "Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with $B^+=1$," J. Differential Geom., vol. 58, iss. 2, pp. 331-370, 2001.
    @article {LL, MRKEY = {1913946},
      AUTHOR = {Li, Tian-Jun and Liu, Ai-Ko},
      TITLE = {Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with {$B\sp +=1$}},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {58},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {331--370},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {57R17 (53D35 57R57)},
      MRNUMBER = {1913946},
      MRREVIEWER = {Stefano Vidussi},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1090348329},
      ZBLNUMBER = {1051.57035},
      }
  • [Mdef] D. McDuff, "From symplectic deformation to isotopy," in Topics in Symplectic $4$-Manifolds, Int. Press, Somerville, MA, 1998, pp. 85-99.
    @incollection {Mdef, MRKEY = {1635697},
      AUTHOR = {McDuff, Dusa},
      TITLE = {From symplectic deformation to isotopy},
      BOOKTITLE = {Topics in Symplectic {$4$}-Manifolds},
      VENUE={{I}rvine, {CA},
      1996},
      SERIES = {First Int. Press Lect. Ser., I},
      PAGES = {85--99},
      PUBLISHER = {Int. Press, Somerville, MA},
      YEAR = {1998},
      MRCLASS = {57R15 (57R52 57R57)},
      MRNUMBER = {1635697},
      MRREVIEWER = {Paolo Lisca},
      ZBLNUMBER = {0928.57018},
      }
  • [M] Go to document D. McDuff, "Symplectic embeddings of 4-dimensional ellipsoids," J. Topol., vol. 2, iss. 1, pp. 1-22, 2009.
    @article {M, MRKEY = {2499436},
      AUTHOR = {McDuff, Dusa},
      TITLE = {Symplectic embeddings of 4-dimensional ellipsoids},
      JOURNAL = {J. Topol.},
      FJOURNAL = {Journal of Topology},
      VOLUME = {2},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {1--22},
      ISSN = {1753-8416},
      MRCLASS = {53D35 (57R17)},
      MRNUMBER = {2499436},
      MRREVIEWER = {S{\'ı}lvia R. Anjos},
      DOI = {10.1112/jtopol/jtn031},
      ZBLNUMBER = {1166.53051},
      }
  • [Mcf] Go to document D. McDuff, "Symplectic embeddings and continued fractions: a survey," Jpn. J. Math., vol. 4, iss. 2, pp. 121-139, 2009.
    @article {Mcf, MRKEY = {2576029},
      AUTHOR = {McDuff, Dusa},
      TITLE = {Symplectic embeddings and continued fractions: a survey},
      JOURNAL = {Jpn. J. Math.},
      FJOURNAL = {Japanese Journal of Mathematics},
      VOLUME = {4},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {121--139},
      ISSN = {0289-2316},
      MRCLASS = {53D35 (11J70)},
      MRNUMBER = {2576029},
      MRREVIEWER = {Tobias Ekholm},
      DOI = {10.1007/s11537-009-0926-9},
      ZBLNUMBER = {1222.53081},
      }
  • [Mc.hofer] Go to document D. McDuff, "The Hofer conjecture on embedding symplectic ellipsoids," J. Differential Geom., vol. 88, iss. 3, pp. 519-532, 2011.
    @article{Mc.hofer,
      author = {McDuff, Dusa},
      TITLE={The {H}ofer conjecture on embedding symplectic ellipsoids},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {88},
      YEAR = {2011},
      NUMBER = {3},
      PAGES = {519--532},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53Dxx},
      MRNUMBER = {2844441},
      ZBLNUMBER = {06011496},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1321366358},
      }
  • [MP] Go to document D. McDuff and L. Polterovich, "Symplectic packings and algebraic geometry," Invent. Math., vol. 115, iss. 3, pp. 405-434, 1994.
    @article {MP, MRKEY = {1262938},
      AUTHOR = {McDuff, Dusa and Polterovich, Leonid},
      TITLE = {Symplectic packings and algebraic geometry},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {115},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {405--434},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {58F05 (14J26 57R40)},
      MRNUMBER = {1262938},
      MRREVIEWER = {Hansj{ö}rg Geiges},
      DOI = {10.1007/BF01231766},
      ZBLNUMBER = {0833.53028},
      }
  • [DoM] D. Müller, Symplectic embeddings of ellipsoids into polydiscs.
    @misc{DoM,
      author={Müller, D.},
      TITLE={Symplectic embeddings of ellipsoids into polydiscs},
      NOTE={Ph.D. thesis, Université de Neuchâtel, in preparation},
     }
  • [Op] Go to document E. Opshtein, "Maximal symplectic packings in $\Bbb P^2$," Compos. Math., vol. 143, iss. 6, pp. 1558-1575, 2007.
    @article {Op, MRKEY = {2371382},
      AUTHOR = {Opshtein, Emmanuel},
      TITLE = {Maximal symplectic packings in {$\Bbb P\sp 2$}},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {143},
      YEAR = {2007},
      NUMBER = {6},
      PAGES = {1558--1575},
      ISSN = {0010-437X},
      MRCLASS = {53D35 (53C55 57R17 57R40)},
      MRNUMBER = {2371382},
      MRREVIEWER = {Hansj{ö}rg Geiges},
      DOI = {10.1112/S0010437X07003041},
      ZBLNUMBER = {1133.53057},
      }
  • [Op2] E. Opshtein, Singular polarizations and symplectic embeddings.
    @misc{Op2,
      author = {Opshtein, Emmanuel},
      TITLE={Singular polarizations and symplectic embeddings},
      ARXIV={1011.6358},
      SORTYEAR={2012},
     }
  • [S] Go to document P. Seidel, "Lectures on four-dimensional Dehn twists," in Symplectic 4-Manifolds and Algebraic Surfaces, New York: Springer-Verlag, 2008, vol. 1938, pp. 231-267.
    @incollection {S, MRKEY = {2441414},
      AUTHOR = {Seidel, Paul},
      TITLE = {Lectures on four-dimensional {D}ehn twists},
      BOOKTITLE = {Symplectic 4-Manifolds and Algebraic Surfaces},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1938},
      PAGES = {231--267},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2008},
      MRCLASS = {53D35 (57R17)},
      MRNUMBER = {2441414},
      DOI = {10.1007/978-3-540-78279-7_4},
      ZBLNUMBER = {1152.53069},
      }
  • [Syl] J. J. Sylvester, "On a remarkable modification of Sturm’s theorem," Philosophical Magazine, vol. V, pp. 446-456, 1853.
    @article{Syl,
      author={Sylvester, J. J.},
      TITLE={On a remarkable modification of {S}turm's theorem},
      JOURNAL={Philosophical Magazine},
      VOLUME={V},
      YEAR={1853},
      PAGES={446--456},
     }
  • [T] Go to document C. H. Taubes, "Embedded contact homology and Seiberg-Witten Floer cohomology I," Geom. Topol., vol. 14, iss. 5, pp. 2497-2581, 2010.
    @article {T, MRKEY = {2746723},
      AUTHOR = {Taubes, Clifford Henry},
      TITLE = {Embedded contact homology and {S}eiberg-{W}itten {F}loer cohomology {I}},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {14},
      YEAR = {2010},
      NUMBER = {5},
      PAGES = {2497--2581},
      ISSN = {1465-3060},
      MRCLASS = {53D42 (53D40 57R57 57R58)},
      MRNUMBER = {2746723},
      MRREVIEWER = {{Ç}a{\u{g}}atay Kutluhan},
      DOI = {10.2140/gt.2010.14.2497},
      ZBLNUMBER = {05841140},
      }

Authors

Dusa McDuff

Department of Mathematics
Barnard College
Columbia University
New York, NY 10027

Felix Schlenk

Université de Neuchâtel
Neuchâtel
Switzerland