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

Abstract

Abstract We consider the embedding function

$$c_b(a)$$

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

$$c_b(a)$$

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

$$b>2$$

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

$$P(\lambda ,\lambda b)$$

P ( λ , λ b )

. Denoted RF(b), this rigid-flexible (RF) value is piecewise smooth with a discrete set of discontinuities for

$$b>2$$

b > 2

. At the same time, by exhibiting a sequence of obstructive classes for

$$b_n = \frac{n+1}{n}$$

b n

=

n + 1

n

at

$$a=8$$

a = 8

, we show that RF is also discontinuous at

$$b=1$$

b = 1

.