Infinite staircases of symplectic embeddings of ellipsoids into Hirzebruch surfaces
Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in 2012, McDuff-Schlenk computed the embedding capacity function of the ball, whose value at a is the size of the smallest four-dimensional ball into which the ellipsoid E(1,a) symplectically embeds. They found that it contains an “infinite staircase” of piecewise-linear sections accumulating from below to the golden ratio to the fourth power. However, infinite staircases seem to be rare for more general targets. Work of Cristofaro-Gardiner-Holm-Mandini-Pires suggests that, up to scaling, there are only finitely many rational symplectic toric manifolds whose embedding capacity functions contain infinite staircases, while Usher has found infinitely many irrational polydisks with infinite staircases. Using ECH capacities in conjunction with the methods of McDuff-Schlenk, we will explain how we have found several infinite families of Hirzebruch surfaces whose embedding capacity functions we expect to contain an infinite staircase. Many of these staircases are “descending” rather than “ascending." This is joint work with Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, and Ana Rita Pires.