Special eccentricities of rational four-dimensional ellipsoids

Abstract

A striking result of McDuff and Schlenk asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional symplectic ball, the answer is governed by an “infinite staircase” determined by the odd-index Fibonacci numbers and the golden mean.

There has recently been considerable interest in better understanding this phenomenon for more general embedding problems. Here we study embeddings of one four-dimensional symplectic ellipsoid into another, and we show that if the target is rational, then the infinite staircase phenomenon found by McDuff and Schlenk can be characterized completely. Specifically, in the rational case, we show that there is an infinite staircase in precisely three cases: when the target has “eccentricity” $1,2$, or $\frac{3}{2}$. In each of these cases, work of Casals and Vianna shows that the corresponding embeddings can be constructed explicitly using polytope mutation; meanwhile, for all other eccentricities, the embedding function is given by the classical volume obstruction, except on finitely many compact intervals, on which it is linear.

Our work verifies in the special case of ellipsoids a conjecture by Holm, Mandini, Pires and the author. The case where the target is the ellipsoid $E(4,3)$ is also interesting from the point of view of this Cristofaro-Gardiner–Holm–Mandini–Pires work: the “staircase obstruction” introduced in that work vanishes for this target, but nevertheless a staircase does not exist. To prove this, we introduce a new combinatorial technique for understanding the obstruction coming from embedded contact homology which is applicable in other situations where the staircase obstruction vanishes, and so is potentially of independent interest.