Equality of Gutt–Hutchings and Ekeland–Hofer capacities

Determine whether, for every integer k ≥ 1 and for all Liouville domains (or other symplectic domains where both capacities are defined), the k-th Gutt–Hutchings capacity c_k^{GH} equals the k-th Ekeland–Hofer capacity c_k^{EH}. Establishing this equality would connect capacities defined via S^1-equivariant symplectic homology with the classical Ekeland–Hofer capacities across all indices k.

Background

Gutt and Hutchings defined a sequence of symplectic capacities c_k^{GH} (k ≥ 1) using S^1-equivariant symplectic homology. The classical sequence of Ekeland–Hofer capacities c_k^{EH} arises from variational methods and has been a central tool in symplectic topology.

The paper compares the first relative Gutt–Hutchings capacity with a relative symplectic (co)homology capacity and proves equality under a dynamical convexity assumption, providing evidence toward relationships between different capacity constructions. Nonetheless, the general equality c_k^{GH} = c_k^{EH} for all k remains conjectural and is explicitly stated as such in the literature.