The embedding capacity of 4-dimensional symplectic ellipsoids - Mathematics > Symplectic Geometry

Abstract: This paper calculates the function $ca$ whose value at $a$ is the infimumof the size of a ball that contains a symplectic image of the ellipsoid$E1,a$. Here $a \ge 1$ is the ratio of the area of the large axis to that ofthe smaller axis. The structure of the graph of $ca$ is surprisingly rich.The volume constraint implies that $ca$ is always greater than or equal tothe square root of $a$, and it is not hard to see that this is equality forlarge $a$. However, for $a$ less than the fourth power $\tau^4$ of the goldenratio, $ca$ is piecewise linear, with graph that alternately lies on a linethrough the origin and is horizontal. We prove this by showing that there areexceptional curves in blow ups of the complex projective plane whose homologyclasses are given by the continued fraction expansions of ratios of Fibonaccinumbers. On the interval $\tau^4,7$ we find $ca=a+1-3$. For $a \ge 7$,the function $ca$ coincides with the square root except on a finite number ofintervals where it is again piecewise linear.The embedding constraints coming from embedded contact homology give rise toanother capacity function $c {ECH}$ which may be computed by counting latticepoints in appropriate right angled triangles. According to Hutchings andTaubes, the functorial properties of embedded contact homology imply that$c {ECH}a \le ca$ for all $a$. We show here that $c {ECH}a \ge ca$ forall $a$.

Author: Dusa McDuff, Felix Schlenk

Source: https://arxiv.org/