Packing numbers of rational ruled four-manifolds
Published in The Journal of Symplectic Geometry, 2011
Recommended citation: Buse, O., Pinsonnault, M., Packing numbers of rational ruled four-manifolds. J. Symplectic Geom. 11 (2013), no.2, 269–316. https://doi.org/10.4310/JSG.2013.v11.n2.a5
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 $\frac{b}{a} \in \mathbf{N}$, 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.