Parametric Gromov width of Liouville domains
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913863183630336 |
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| author | Broćić, Filip Cant, Dylan |
| author_facet | Broćić, Filip Cant, Dylan |
| contents | The classical Gromov width measures the largest symplectic ball embeddable into a symplectic manifold; inspired by the symplectic camel problem, we generalize this to ask how large a symplectic ball can be embedded as a family over a parameter space $N$. Given a smooth map $f: N \to Ω$, where $Ω$ is a symplectic manifold, we define the \emph{parametric Gromov width} $\mathrm{Gr}(f,Ω)$ as the supremum of capacities $a>0$ for which there exists a family of balls, parametrized by $N$, of capacity $a$ whose centers trace out the map $f$. For Liouville domains $Ω$, we establish upper bounds on $\mathrm{Gr}(f,Ω)$ using the Floer cohomology persistence module associated to $Ω$. Specializing to fiberwise starshaped domains in the cotangent bundle $T^*M$, we derive computable bounds via filtered string topology. Specific examples of $Ω$ -- including disk cotangent bundles of thin ellipsoids, open books, and tori -- demonstrate our bounds, and reveal constraints on parameterized symplectic embeddings beyond the classical Gromov width. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parametric Gromov width of Liouville domains Broćić, Filip Cant, Dylan Symplectic Geometry 53D40, 53D35, 53D05, 55P50 The classical Gromov width measures the largest symplectic ball embeddable into a symplectic manifold; inspired by the symplectic camel problem, we generalize this to ask how large a symplectic ball can be embedded as a family over a parameter space $N$. Given a smooth map $f: N \to Ω$, where $Ω$ is a symplectic manifold, we define the \emph{parametric Gromov width} $\mathrm{Gr}(f,Ω)$ as the supremum of capacities $a>0$ for which there exists a family of balls, parametrized by $N$, of capacity $a$ whose centers trace out the map $f$. For Liouville domains $Ω$, we establish upper bounds on $\mathrm{Gr}(f,Ω)$ using the Floer cohomology persistence module associated to $Ω$. Specializing to fiberwise starshaped domains in the cotangent bundle $T^*M$, we derive computable bounds via filtered string topology. Specific examples of $Ω$ -- including disk cotangent bundles of thin ellipsoids, open books, and tori -- demonstrate our bounds, and reveal constraints on parameterized symplectic embeddings beyond the classical Gromov width. |
| title | Parametric Gromov width of Liouville domains |
| topic | Symplectic Geometry 53D40, 53D35, 53D05, 55P50 |
| url | https://arxiv.org/abs/2504.15207 |