Symplectic embeddings of $4$-dim ellipsoids into cubes

Recently, McDuff and Schlenk determined in [MS] the function c EB (a) whose value at a is the infimum of the size of a 4-ball into which the ellipsoid E(1, a) symplectically embeds (here, a 1 is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid). In this paper we look at embeddings into four-dimensional cubes instead, and determine the function c EC (a) whose value at a is the infimum of the size of a 4-cube C 4 (A) = D 2 (A) D 2 (A) into which the ellipsoid E(1, a) symplectically embeds (where D 2 (A) denotes the disc in R 2 of area A). As in the case of embeddings into balls, the structure of the graph of c EC (a) is very rich: for a less than the square 2 of the silver ratio := 1 + 2, the function c EC (a) turns out to be piecewise linear, with an infinite staircase converging to ( 2 , 2 /2). This staircase is determined by Pell numbers. On the interval 2 , 7 1 32 , the function c EC (a) coincides with the volume constraint a 2 except on seven disjoint intervals, where c is piecewise linear. Finally, for a 7 1 32 , the functions c EC (a) and a 2 are equal. For the proof, we first translate the embedding problem E(1, a) C 4 (A) to a certain ball packing problem of the ball B 4 (2A). This embedding problem is then solved by adapting the method from [MS], which finds all exceptional spheres in blow-ups of the complex projective plane that provide an embedding obstruction.