Symplectic embeddings of 4-dimensional ellipsoids into cubes

TL;DR

This paper characterizes the symplectic embedding capacity of 4-dimensional ellipsoids into cubes, revealing a complex structure with infinite staircases related to Pell numbers, extending previous work on embeddings into balls.

Contribution

It determines the embedding function c_{EC}(a) for ellipsoids into cubes, showing its detailed structure and relation to volume constraints, and connects it to embeddings into ellipsoids E(A,2A).

Findings

The embedding function c_{EC}(a) is piecewise linear below sigma^2 and follows a staircase pattern.

For a >= 7+1/32, c_{EC}(a) equals the volume constraint sqrt(a/2).

The structure of c_{EC}(a) involves Pell numbers and converges to specific points.

Abstract

Recently, McDuff and Schlenk determined 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) 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 c_{EC}(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+1/32], the function c_{EC}(a) coincides with the volume constraint sqrt(a/2) except on seven disjoint intervals, where c is piecewise linear. Finally, for a >= 7+1/32, the functions c_{EC}(a) and sqrt(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 McDuff-Schlenk, 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 elllipsoid 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.