The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs
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Author(s)
Jin, Alvin
Lee, Andrew
Date Issued
August 24, 2023
Publisher
Springer International Publishing
Citation
Journal of Fixed Point Theory and Applications. 2023 Aug 24;25(3):79
Version
Author's final manuscript
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
.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11784-023-01080-w
