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, $a \geqslant 1$ 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 $C^4(A) = D^2(A) \times D^2(A)$ into which the ellipsoid $E(1, a)$ symplectically embeds (where $D^2(A)$ denotes the disc in $\mathbb{R}^2$ 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 $\sigma^2$ of the silver ratio $\sigma := 1 + \sqrt{2}$, the function cEC(a) turns out to be piecewise linear, with an infinite staircase converging to $(\sigma^2 , \sqrt{\sigma^2 / 2})$. This staircase is determined by Pell numbers. On the interval $[\sigma^2 , 7 \frac{1}{32}]$, the function $c_{EC}(a)$ coincides with the volume constraint $\sqrt{\frac{a}{2}}$ except on seven disjoint intervals, where $c$ is piecewise linear. Finally, for $a \geqslant 7 \frac{1}{32}$, the functions $c_{EC}(a)$ and $\sqrt{\frac{a}{2}}$ are equal.

For the proof, we first translate the embedding problem $E(1, a)$ $\hookrightarrow C^4(A)$ 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.

We also prove that the ellipsoid $E(1, a)$ symplectically embeds into the cube $C^4(A)$ if and only if $E(1, a)$ symplectically embeds into the ellipsoid $E(A, 2A)$. Our embedding function $c_{EC}(a)$ thus also describes the smallest dilate of $E(1, 2)$ into which $E(1, a)$ symplectically embeds.

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Published 17 March 2016