Packing numbers of rational ruled 4-manifolds

TL;DR

This paper completely solves the symplectic packing problem for rational, ruled 4-manifolds, providing explicit formulas for packing numbers and related invariants, and clarifies embedding conditions for ellipsoids into polydisks.

Contribution

It offers explicit formulas for packing numbers, Gromov widths, and obstructing classes for rational ruled 4-manifolds, advancing understanding of symplectic embeddings.

Findings

Explicit formulas for packing numbers and Gromov widths.

Determination of embedding conditions for ellipsoids into polydisks.

Alternative proof of sharpness of ECH capacities for ellipsoid embeddings.

Abstract

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding ellipsoids into polydisks.