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Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids

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Abstract

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks P(a, 1) into four-dimensional ellipsoids E(bc, c) when \(1\le a< 2\) and b is a half-integer. When \(1 \le a < 2-O(b^{-1})\) we demonstrate that P(a, 1) symplectically embeds into E(bc, c) if and only if \(a+b\le bc\). Our results show that inclusion is optimal and extend the result by Hutchings (Geom Topol 20(2):1085–1126, 2016) when b is an integer. Our proof is based on a combinatorial criterion developed by Hutchings [14] to obstruct symplectic embeddings.

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Notes

  1. Note that the terminology “convex” in [4] is slightly broader than ours.

  2. Given a cooriented contact manifold \((M,\text{ ker }\lambda )\), the Reeb vector field R is uniquely determined by the equations \(\lambda (R)=1\) and \(d\lambda (R, \cdot ) =0\).

  3. Here we used the ECH capacities of a ball ([12, Cor. 1.3]) and Thm. 1.17 to obtain this computation.

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Acknowledgements

We would like to thank Michael Hutchings for suggesting this project and for helpful discussions. We thank the referee for their careful reading and suggestions on how to clarify the results of this paper. Our 2020 Virtual BeECH Group was supported by NSF Grant DMS-1840723. Additionally, LD was partially supported by NSF Grants DMS-1745670, DMS-1840723, DMS-2104411; JN was partially supported by NSF Grants DMS-1840723, DMS-2104411; and MW was partially supported by NSF Grants DMS-1745670, DMS-2103245, DMS-1810692.

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Authors and Affiliations

  1. Rice University, Houston, USA

    L. Digiosia & J. Nelson

  2. University of Washington, Seattle, USA

    H. Ning & Y. Yang

  3. Cornell University, Ithaca, USA

    M. Weiler

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  1. L. Digiosia
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  2. J. Nelson
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  3. H. Ning
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  4. M. Weiler
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  5. Y. Yang
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Correspondence to J. Nelson.

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Digiosia, L., Nelson, J., Ning, H. et al. Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids. J. Fixed Point Theory Appl. 24, 69 (2022). https://doi.org/10.1007/s11784-022-00981-6

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