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 $a$.