Hopf orbits and the first ECH capacity

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 . 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 embedded contact homology (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.