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On certain quantifications of Gromov's nonsqueezing theorem

Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J Zhu

Appendix: Joé Brendel

Geometry & Topology 28 (2024) 1113–1152
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 × 2. By Gromov’s nonsqueezing theorem, E must be nonempty. 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 2. In the appendix by Joé Brendel, it is shown that the lower bound is optimal for R < 3. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

Keywords
Gromov nonsqueezing, waist inequality, folding
Mathematical Subject Classification
Primary: 53D05, 53D35
References
Publication
Received: 22 May 2021
Revised: 5 April 2022
Accepted: 11 September 2022
Published: 10 May 2024
Proposed: Yakov Eliashberg
Seconded: Leonid Polterovich, Tobias H Colding
Authors
Kevin Sackel
Department of Mathematics and Statistics
University of Massachusetts, Amherst
Amherst, MA
United States
Antoine Song
Mathematics Department
California Institute of Technology
Pasadena, CA
United States
Umut Varolgunes
Department of Mathematics
Boğaziçi University
İstanbul
Turkey
Jonathan J Zhu
Department of Mathematics
University of Washington
Seattle, WA
United States
Joé Brendel
Institut de Mathématiques
Université de Neuchâtel
Neuchâtel
Switzerland

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