Journal of Symplectic Geometry

Volume 11 (2013)

Number 2

Packing numbers of rational ruled four-manifolds

Pages: 269 – 316



Olguta Buse (Department of Mathematical Sciences, IUPUI, Indianapolis, Ind., U.S.A.)

Martin Pinsonnault (Department of Mathematics, The University of Western Ontario, London, Ontario, Canada)


We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic four-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 $\frac{b}{a} \in \mathbb{N}$, 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 embedded contact homology capacities give sharp inequalities for embedding ellipsoids into polydisks.

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