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Higher symplectic capacities and the stabilized embedding problem for integral elllipsoids

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Abstract

The third named author has been developing a theory of “higher” symplectic capacities. These capacities are invariant under taking products, and so are well suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.

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Notes

  1. Actually, only the \(N = 0\) case of these functions was defined, but the definition extends verbatim to general N, and that will be our working definition here.

  2. In fact, Proposition 1.6 likely describes the entirety of the first step, although we do not address this here.

  3. Strictly speaking, \(X \times B^2(S)\) is not a Liouville domain, since it has corners, although these can be removed by an arbitrarily small smoothing. See [28, §5.4] for a more precise formulation. Property (1) is of course automatic given property (3).

  4. There is also a nice story extending the theory to non-exact symplectic cobordisms, but we will ignore this for simplicity.

  5. More precisely, we only allow “good” Reeb orbits, and we count cylinders which are additionally “anchored” in X.

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Acknowledgements

We thank Felix Schlenk for his encouragement, and for helping the first and third named authors better understand constructions of embeddings between stabilized ellipsoids. We would also like to thank the referee for carefully reading our paper and for many useful comments. Our paper is dedicated to Claude Viterbo on the occasion of his 60th birthday. We are immensely grateful to Claude for his visionary leadership of our field. This research was completed, while the first named author was on a von Neumann fellowship at the Institute for Advanced Study; he thanks the Institute for their support. The first named author is partially supported by NSF grant DMS-1711976 and the second named author by Simons Foundation Grant no. 633715.

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Correspondence to Daniel Cristofaro-Gardiner.

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This article is part of the topical collection “Symplectic geometry—A Festschrift in honour of Claude Viterbo’s 60th birthday” edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk.

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Cristofaro-Gardiner, D., Hind, R. & Siegel, K. Higher symplectic capacities and the stabilized embedding problem for integral elllipsoids. J. Fixed Point Theory Appl. 24, 49 (2022). https://doi.org/10.1007/s11784-022-00942-z

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