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On certain quantifications of Gromov's non-squeezing theorem

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

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 $R \leq \sqrt{2}$. 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.

On certain quantifications of Gromov's non-squeezing theorem

Abstract

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

Paper Structure

This paper contains 23 sections, 27 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Minkowski dimension problem
  3. The Lipschitz problem
  4. Preliminaries
  5. Waist inequalities
  6. Tubes around minimal submanifolds
  7. Gromov foliation and maps into the cylinder
  8. A quantitative obstruction to partial symplectic embeddings
  9. Squeezing the complement of a Lagrangian plane
  10. Proof of Proposition \ref{['extension']}
  11. Lipschitz symplectic embeddings of balls
  12. Further questions
  13. Minkowski dimension problem in higher dimensions
  14. Capacity after removing a linear plane
  15. Minkowski dimension problem for large R
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists a symplectic embedding of $B^{2n}(\pi R^2)$ into $Z^{2n}(\pi r^2)$ if and only if $R \leq r$.

Figures (2)

  • Figure 1: We stretch the region between $X_i$ and $X_{i+1}$ in an area-preserving way so that the images $Y_i$ and $Y_{i+1}$ are separated by distance just over $1$.
  • Figure 2: The triangle $\Delta_{a,b,c}$ as union $W \cup V_a \cup V_b \cup V_c$, on the left as the toric moment polytope of $\operatorname{\mathbb{CP}}(a^2,b^2,c^2)$ and on the right as almost-toric base diagram of $\operatorname{\mathbb{CP}}^2$. In both cases the fibration is toric over $W$ and lens spaces fibre over the segments $\ell_a,\ell_b,\ell_c$.

Theorems & Definitions (55)

  • Theorem 1.1: Gromov's Nonsqueezing Theorem Gro85
  • Theorem 1.2: Katok Kat73
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 2.1: Waist inequality
  • Theorem 2.2: Heintze-Karcher inequality
  • ...and 45 more