Symplectic embeddings of 4-dimensional ellipsoids

We show how to reduce the problem of symplectically embeddingone 4-dimensional rational ellipsoid into another to a problemof embedding disjoint unions of balls into P². For example,the problem of embedding the ellipsoid E(1, k) into a ball Bis equivalent to that of embedding k disjoint equal balls intoP², and so can be solved by the work of Gromov, McDuff–Polterovich,and Biran. (Here k is the ratio of the area of the major axisto that of the minor axis.) As a consequence we show that theball may be fully filled by the ellipsoid E(1, k) for k = 1,4 and all k 9, thus answering a question raised by Hofer. Received March 31, 2008.