Rigidity and flexibility in $p$-adic symplectic geometry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913832865103872 |
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| author | Crespo, Luis Pelayo, Álvaro |
| author_facet | Crespo, Luis Pelayo, Álvaro |
| contents | Let $n\ge 2$ be an integer and let $p$ be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for $p$-adic embeddings: for any $p$-adic absolute value $R$, the entire $p$-adic space $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to the $p$-adic cylinder $\mathrm{Z}_p^{2n}(R)$ of radius $R$, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the $p$-adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the $p$-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant $p$-adic analytic symplectic capacities, of which the $p$-adic equivariant Gromov width is an example. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity and flexibility in $p$-adic symplectic geometry Crespo, Luis Pelayo, Álvaro Symplectic Geometry Mathematical Physics 53D35, 37P05 Let $n\ge 2$ be an integer and let $p$ be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for $p$-adic embeddings: for any $p$-adic absolute value $R$, the entire $p$-adic space $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to the $p$-adic cylinder $\mathrm{Z}_p^{2n}(R)$ of radius $R$, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the $p$-adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the $p$-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant $p$-adic analytic symplectic capacities, of which the $p$-adic equivariant Gromov width is an example. |
| title | Rigidity and flexibility in $p$-adic symplectic geometry |
| topic | Symplectic Geometry Mathematical Physics 53D35, 37P05 |
| url | https://arxiv.org/abs/2505.07663 |