On certain quantifications of Gromov's non-squeezing theorem

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 ${\mathbb{D}}^2\times {\mathbb{R}}^2$. 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 $R\le \sqrt{2}$. In an appendix by Joé Brendel, it is shown that the lower bound is optimal for $R<\sqrt{3}$. We also discuss the minimum volume of $E$ in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

Revision includes an appendix by J. Brendel. Version accepted in Geometry & Topology

References in corpus (2)

• Markov numbers and Lagrangian cell complexes in the complex projective plane
• A tight estimate for the waist of the ball

Cited by in corpus (1)

• Quantitative Results on Symplectic Barriers