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Published May 3, 2021 | public
Discussion Paper Metadata-only

On certain quantifications of Gromov's non-squeezing theorem

Sackel, Kevin
Song, Antoine
Varolgunes, Umut
Zhu, Jonathan J.
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Abstract

Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D² × R². By Gromov's non-squeezing theorem, E must be non-empty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R ≤ √2̅. In an appendix by Joé Brendel, it is shown that the lower bound is optimal for R < √3̅. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

Additional Information

Revision includes an appendix by J. Brendel. Version accepted in Geometry & Topology. We thank Michael Usher for a useful e-mail correspondence. We also thank Felix Schlenk and Jo´e Brendel for their interest and helpful comments our paper. K.S. thanks Larry Guth for originally suggesting the version of this problem involving Lipschitz constants which provided the initial impetus for this project. U.V. thanks Grigory Mikhalkin for very useful discussions on Theorem 1.3 and also sketching a proof of Corollary 6.5 that we ended up not using; and Kyler Siegel for a discussion regarding Section 6.2. K.S. was partially supported by the National Science Foundation under grant DMS-1547145. This research was conducted during the period A.S. served as a Clay Research Fellow. J.Z. was supported in part by the National Science Foundation under grant DMS-1802984 and the Australian Research Council under grant FL150100126.

Additional details

Identifiers Funding Dates
Created:
August 20, 2023
Modified:
February 1, 2025