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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 thekth Ekeland–Hofer capacity for all ellipsoids can characterize it on a class of domains. We prove that fork=n=2, this holds for convex toric domains, but not for all monotone toric domains. We also prove that, fork=n\ge 3, this does not hold even for convex toric domains.
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This content will become publicly available on July 25, 2026
The Ruelle invariant and convexity in higher dimensions
We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domainXin\mathbb{R}^{2n}, we prove that the Ruelle invariant\operatorname{Ru}(X), the period of the systolec(X)and the volume\operatorname{vol}(X)satisfy\operatorname{Ru}(X) \cdot c(X) \le C(n) \cdot \operatorname{vol}(X). HereC(n) > 0is an explicit constant depending onn. As an application, we construct dynamically convex contact forms onS^{2n-1}that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension.
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- Award ID(s):
- 2103165
- PAR ID:
- 10656717
- Publisher / Repository:
- Journal of the European Mathematics Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- ISSN:
- 1435-9855
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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