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The embedding capacity of 4-dimensional symplectic ellipsoids. (English) Zbl 1254.53111

Let \(E(1, a)\) denote the 4-dimensional ellipsoid with the ratio \(a\) of the area of the major axis to that of the minor axis. The authors calculate the function \(c(a)=\inf\{\mu \mid E(1, a)\overset {s} \hookrightarrow B(\mu)\}\), where \(B(\mu)\) is the open ball with radius \(\mu\) and \(s\) means a symplectic embedding. It is shown that the structure of the graph of \(c(a)\) surprisingly rich. The volume constraint implies that \(c(a)\) is greater than or equal to the square root of \(a\), and that this is an equality for large \(a\). However, for \(a\) less than the fourth power \(\tau^4\) of the golden ratio \(\tau\), \(c(a)\) is piecewise linear. This is proved by showing that there are exceptional curves in blow ups of the complex projective plane where homology classes are given by the continued fraction expansions of rations of Fibonacci numbers. On the interval \([\tau^4, 7]\), the authors find that \(c(a)=(a+1)/3\) and, for \(a>7\), the function \(c(a)\) coincides with the square root except on a finite number of intervals where it is again piecewise linear. In the arguments to compute \(c(a)\) on the intertval \([7, 8]\), a computer is used.
The embedding constraints coming from embedding contact homology give rise to another capacity function \(c_{ECH}\) which may be computed by counting lattice points in appropriate right-angled triangles by using Fibonacci numbers. According to M. Hatchings and C. H. Taubes [J. Differ. Geom. 88, No. 2, 231–266 (2011; Zbl 1238.53061); J. Symplectic Geom. 5, No. 1, 43–137 (2007; Zbl 1157.53047)], the functional properties of embedded contact homology imply that \(c_{\mathrm{ECH}}(a)\leq c(a)\) for all \(a\). The authors shows in this paper that \(c_{\mathrm{ECH}}(a)\geq c(a)\) for all \(a\).

MSC:

53D35 Global theory of symplectic and contact manifolds
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

[1] P. Biran, ”Symplectic packing in dimension \(4\),” Geom. Funct. Anal., vol. 7, iss. 3, pp. 420-437, 1997. · Zbl 0892.53022 · doi:10.1007/s000390050014
[2] P. Biran, ”Constructing new ample divisors out of old ones,” Duke Math. J., vol. 98, iss. 1, pp. 113-135, 1999. · Zbl 0961.14005 · doi:10.1215/S0012-7094-99-09803-4
[3] 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. · Zbl 1047.53054
[4] O. Buse and R. Hind, ”Symplectic embeddings of ellipsoids in dimension greater than four,” Geom. and Top., vol. 15, pp. 2091-2110, 2011. · Zbl 1239.53107 · doi:10.2140/gt.2011.15.2091
[5] 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. · Zbl 1143.53341 · doi:10.1017/CBO9780511755187.002
[6] I. Ekeland and H. Hofer, ”Symplectic topology and Hamiltonian dynamics. II,” Math. Z., vol. 203, iss. 4, pp. 553-567, 1990. · Zbl 0729.53039 · doi:10.1007/BF02570756
[7] L. Guth, ”Symplectic embeddings of polydisks,” Invent. Math., vol. 172, iss. 3, pp. 477-489, 2008. · Zbl 1153.53060 · doi:10.1007/s00222-007-0103-9
[8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Third ed., Oxford, at the Clarendon Press, 1954. · Zbl 0058.03301
[9] R. Hind and E. Kerman, New obstructions to symplectic embeddings. · Zbl 1296.53160 · doi:10.1007/s00222-013-0471-2
[10] M. Hutchings, ”Quantitative embedded contact homology,” J. Differential Geom., vol. 88, iss. 2, pp. 231-266, 2011. · Zbl 1238.53061
[11] M. Hutchings, ”Recent progress on symplectic embedding problems in four dimensions,” Proc. Natl. Acad. Sci. USA, vol. 108, iss. 20, pp. 8093-8099, 2011. · Zbl 1256.53054 · doi:10.1073/pnas.1018622108
[12] 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. · Zbl 1157.53047 · doi:10.4310/JSG.2007.v5.n1.a5
[13] B. Li and T. Li, ”Symplectic genus, minimal genus and diffeomorphisms,” Asian J. Math., vol. 6, iss. 1, pp. 123-144, 2002. · Zbl 1008.57024
[14] 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. · Zbl 1051.57035
[15] D. McDuff, ”From symplectic deformation to isotopy,” in Topics in Symplectic \(4\)-Manifolds, Int. Press, Somerville, MA, 1998, pp. 85-99. · Zbl 0928.57018
[16] D. McDuff, ”Symplectic embeddings of 4-dimensional ellipsoids,” J. Topol., vol. 2, iss. 1, pp. 1-22, 2009. · Zbl 1166.53051 · doi:10.1112/jtopol/jtn031
[17] D. McDuff, ”Symplectic embeddings and continued fractions: a survey,” Jpn. J. Math., vol. 4, iss. 2, pp. 121-139, 2009. · Zbl 1222.53081 · doi:10.1007/s11537-009-0926-9
[18] D. McDuff, ”The Hofer conjecture on embedding symplectic ellipsoids,” J. Differential Geom., vol. 88, iss. 3, pp. 519-532, 2011. · Zbl 1239.53109
[19] D. McDuff and L. Polterovich, ”Symplectic packings and algebraic geometry,” Invent. Math., vol. 115, iss. 3, pp. 405-434, 1994. · Zbl 0833.53028 · doi:10.1007/BF01231766
[20] D. Müller, Symplectic embeddings of ellipsoids into polydiscs.
[21] E. Opshtein, ”Maximal symplectic packings in \(\mathbb P^2\),” Compos. Math., vol. 143, iss. 6, pp. 1558-1575, 2007. · Zbl 1133.53057 · doi:10.1112/S0010437X07003041
[22] E. Opshtein, Singular polarizations and symplectic embeddings. · Zbl 1405.53119
[23] 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. · Zbl 1152.53069 · doi:10.1007/978-3-540-78279-7_4
[24] J. J. Sylvester, ”On a remarkable modification of Sturm’s theorem,” Philosophical Magazine, vol. V, pp. 446-456, 1853.
[25] C. H. Taubes, ”Embedded contact homology and Seiberg-Witten Floer cohomology I,” Geom. Topol., vol. 14, iss. 5, pp. 2497-2581, 2010. · Zbl 1275.57037 · doi:10.2140/gt.2010.14.2497
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