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

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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 RF is also discontinuous at \(b=1\).

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Acknowledgements

We thank Dan Cristofaro-Gardiner for suggesting this problem, and for both his and Felix Schlenk’s patience in explaining the arguments in [3]. We also thank the anonymous referee for their feedback which greatly improved this work.

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Correspondence to Andrew Lee.

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Jin, A., Lee, A. The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs. J. Fixed Point Theory Appl. 25, 79 (2023). https://doi.org/10.1007/s11784-023-01080-w

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