On symplectic capacities and their blind spots

Abstract

In this paper, we settle three basic questions concerning the Gutt–Hutchings capacities from [11] which are conjecturally equal to the Ekeland–Hofer capacities from [7, 8]. Our primary result settles a version of the recognition question from [4], in the negative. We prove that the Gutt–Hutchings capacities together with the volume do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. In particular, we construct a smooth family of such domains, all of which have the same Gutt–Hutchings capacities and volume, but no two of which are symplectomorphic. We also establish two independence properties of the Gutt–Hutchings capacities. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume by constructing a family of star-shaped domains with smooth boundaries, whose Gutt–Hutchings capacities all agree but whose volumes all differ. We also prove that the capacities are mutually independent by constructing, for any [Formula: see text], a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the [Formula: see text]th. The constructions underlying these results are not exotic. They are convex and concave toric domains as defined in [11], where the authors also establish beautiful formulae for their capacities. A key to the progress made here is a significant simplification of these formulae under an additional symmetry assumption. This simplification allows us to identify new blind spots of the Gutt–Hutchings capacities which are used to construct the desired examples. This simplification also yields explicit computations of the Gutt–Hutchings capacities in many new examples.