The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs

Abstract

We consider the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$. Previous work suggests that determining the entirety of $c_b(a)$ for all $b$ is difficult, as infinite staircases can appear for many sequences of irrational $b$. In contrast, we show that for every polydisc $P(1,b)$ with $b>2$, there is an explicit formula for the minimum $a$ such that the embedding problem is determined only by volume. That is, when the ellipsoid is sufficiently stretched, there is a symplectic embedding of $E(1,a)$ fully filling an appropriately scaled polydisc $P(\lambda ,\lambda b)$. Denoted $RF(b)$, this rigid-flexible ($RF$) value is piecewise smooth with a discrete set of discontinuities for $b>2$. At the same time, by exhibiting a sequence of obstructive classes for $b_n=\frac{n+1}{n}$ at $a=8$, we show % that $c_{b_n}(8)$ is above the volume constraint. So, in combination with the Frenkel-Müller result, it follows that $RF$ is also discontinuous at $b=1$.

40 pages, 1 figure