Rigid-flexible values for symplectic embeddings of four-dimensional ellipsoids into almost-cubes

Abstract

We consider the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest possible scaling by $\lambda >1$ of the polydisc $P(1,b)$. In particular, we calculate rigid-flexible values, i.e. the minimum $a$ such that for $E(1,a^{\prime })$ with $a^{\prime }>a$, the embedding problem is determined only by volume. For $1<b<2$ we find that these values vary piecewise smoothly outside the discrete set $b\in {\left(\frac{n+1}{n}\right)}^2$. As Jin and Lee analyze packing stability in the caes $b>2$, our results complete the story outside of a discrete set.