On certain quantifications of Gromov's non-squeezing theorem
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914803372523520 |
|---|---|
| author | Sackel, Kevin Song, Antoine Varolgunes, Umut Zhu, Jonathan J. |
| author_facet | Sackel, Kevin Song, Antoine Varolgunes, Umut Zhu, Jonathan J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_00586 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On certain quantifications of Gromov's non-squeezing theorem Sackel, Kevin Song, Antoine Varolgunes, Umut Zhu, Jonathan J. Symplectic Geometry Differential Geometry 53D05, 53D35 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. |
| title | On certain quantifications of Gromov's non-squeezing theorem |
| topic | Symplectic Geometry Differential Geometry 53D05, 53D35 |
| url | https://arxiv.org/abs/2105.00586 |