The embedding capacity of 4-dimensional symplectic ellipsoids

By D. Mcduff, F. Schlenk. 2009

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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 $c_{ECH}$ 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 $c_{ECH}(a) \le c(a)$ for all $a$. We show here that $c_{ECH}(a) \ge c(a)$ for all $a$.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f774f5e0ff8f256d2b5f8df50bdb687326573b99

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Citations in the corpus

Lecture Notes on Embedded Contact Homology (2013, unknown): “2 (McDuff-Schlenk [44]).”
The lens space realization problem (2010, background): “The use of weight expansions draws inspiration from [33].”
Symplectic embeddings from concave toric domains into convex ones (2014, unknown): “This is to be compared with the case of embedding an ellipsoid into a ball, where the ball packings that arise are much more constrained, see [17].”
Symplectic embeddings into four‐dimensional concave toric domains (2013, background): “For example, an ellipsoid E(a, 1) may embed into a ball B(c) of slightly greater volume, and this is always possible when a ((17) 6 )(2); see [13]; if we set c = λb, then the ellipsoid is not a subset of Z(λ, λb) if we choose b sufficiently large.”
Quantitative embedded contact homology (2010, background): “ embeds into the ball B(c). By Theorem 1.1, Proposition 1.2, and Corollary 1.3, we have f(a) ≥ sup k=2,3,... (a,1)k (1,1)k = sup d=1,2,... 1 d (a,1)(d2+3d+2)/2. (1.4) On the other hand, McDuﬀ-Schlenk [19] computed the function f explicitly, obtaining a beautiful and complicated answer involving Fibo”
Beyond ECH capacities (2014, background): “(It is still a subtle number-theoretic problem to decide whether this embedding criterion holds for any given a ; b ; c ; d ; see eg McDuff and Schlenk [19] and Cristofaro-Gardiner and Kleinman [4].)”
Lagrangian spheres, symplectic surfaces and the symplectic mapping class group (2010, methodology): “rical classes is also valid for exceptional classes. In this case, one gets an explicit K0-equivalence from an exceptional class to Ei when n≥ 3 or possibly H− E1 − E2 when n= 2. This is also used in [46]. 4.1.3 (K,α)−null spherical classes and DK,α(M) In this section we ﬁx a class αin the K−symplec”
Symplectic embedding problems, old and new (2017, background): “to certain Diophantine systems that correspond to special holomorphic spheres in blowups of the complex projective plane ([84] and §8.”