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arXiv · 2312.11830

Hopf orbits and the first ECH capacity

Umberto Hryniewicz·Michael Hutchings·Vinicius G. B. Ramos
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Abstract

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.

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Umberto Hryniewicz, Michael Hutchings, Vinicius G. B. Ramos. 2023-12-19. Hopf orbits and the first ECH capacity. https://arxiv.org/abs/2312.11830

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