Hopf orbits and the first ECH capacity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915509949169664 |
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| author | Hryniewicz, Umberto Hutchings, Michael Ramos, Vinicius G. B. |
| author_facet | Hryniewicz, Umberto Hutchings, Michael Ramos, Vinicius G. B. |
| contents | We consider dynamically convex star-shaped domains in a symplectic vector space of dimension $4$. For such a domain, a ``Hopf orbit'' is a closed characteristic in the boundary which is unknotted and has self-linking number $-1$. We show that the minimum action among Hopf orbits exists and defines a symplectic capacity for dynamically convex star-shaped domains. We further show that this capacity agrees with the first ECH capacity for such domains. Combined with a result of Edtmair, this implies that for dynamically convex star-shaped domains in four dimensions, the first ECH capacity agrees with the cylinder capacity. This also provides a method to show that the first ECH capacity of a dynamically convex star-shaped domain satisfies the axioms of a normalized symplectic capacity without any need for Seiberg-Witten theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11830 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hopf orbits and the first ECH capacity Hryniewicz, Umberto Hutchings, Michael Ramos, Vinicius G. B. Symplectic Geometry Dynamical Systems We consider dynamically convex star-shaped domains in a symplectic vector space of dimension $4$. For such a domain, a ``Hopf orbit'' is a closed characteristic in the boundary which is unknotted and has self-linking number $-1$. We show that the minimum action among Hopf orbits exists and defines a symplectic capacity for dynamically convex star-shaped domains. We further show that this capacity agrees with the first ECH capacity for such domains. Combined with a result of Edtmair, this implies that for dynamically convex star-shaped domains in four dimensions, the first ECH capacity agrees with the cylinder capacity. This also provides a method to show that the first ECH capacity of a dynamically convex star-shaped domain satisfies the axioms of a normalized symplectic capacity without any need for Seiberg-Witten theory. |
| title | Hopf orbits and the first ECH capacity |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2312.11830 |