Hopf orbits and the first ECH capacity

Journal of Topology, vol. 19

Abstract

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.

Authors 3

1. Umberto Hryniewicz Aachen Chair for Geometry and Analysis

   RWTH Aachen University

   Affiliation as printed

   Chair for Geometry and Analysis RWTH Aachen University Aachen Germany

2. Michael Hutchings corresponding

   University of California, Berkeley

   Affiliation as printed

   Department of Mathematics University of California, Berkeley Berkeley California USA

3. Vinicius G. B. Ramos

   Instituto Nacional de Matemática Pura e Aplicada

   Affiliation as printed

   Instituto de Matematica Pura e Aplicada Rio de Janeiro Brazil