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 \(a\).


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
Full Text: DOI arXiv


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