Characterizing symplectic capacities on ellipsoids
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910593094516736 |
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| author | Gutt, Jean Ramos, Vinicius G. B. |
| author_facet | Gutt, Jean Ramos, Vinicius G. B. |
| contents | It is a long-standing conjecture that all symplectic capacities which are equal to the Gromov width for ellipsoids coincide on a class of convex domains in $\mathbb{R}^{2n}$. It is known that they coincide for monotone toric domains in all dimensions. In this paper, we study whether requiring a capacity to be equal to the $k^{th}$ Ekeland-Hofer capacity for all ellipsoids can characterize it on a class of domains. We prove that for $k=n=2$, this holds for convex toric domains, but not for all monotone toric domains. We also prove that for $k=n\ge 3$, this does not hold even for convex toric domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_06476 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterizing symplectic capacities on ellipsoids Gutt, Jean Ramos, Vinicius G. B. Symplectic Geometry 53D05 It is a long-standing conjecture that all symplectic capacities which are equal to the Gromov width for ellipsoids coincide on a class of convex domains in $\mathbb{R}^{2n}$. It is known that they coincide for monotone toric domains in all dimensions. In this paper, we study whether requiring a capacity to be equal to the $k^{th}$ Ekeland-Hofer capacity for all ellipsoids can characterize it on a class of domains. We prove that for $k=n=2$, this holds for convex toric domains, but not for all monotone toric domains. We also prove that for $k=n\ge 3$, this does not hold even for convex toric domains. |
| title | Characterizing symplectic capacities on ellipsoids |
| topic | Symplectic Geometry 53D05 |
| url | https://arxiv.org/abs/2312.06476 |