Coisotropic Ekeland-Hofer capacities

TL;DR

This paper introduces coisotropic Ekeland-Hofer capacities in symplectic geometry, providing new capacity definitions, formulas for convex domains, and relating capacities to leafwise chords on hypersurfaces.

Contribution

It develops relative capacities with respect to coisotropic subspaces, offers representation formulas, and establishes a product formula linking capacities to leafwise chords.

Findings

Defined coisotropic Ekeland-Hofer capacities.

Derived representation formulas for convex domains.

Connected capacities to leafwise chords on hypersurfaces.

Abstract

For subsets in the standard symplectic space $(\mathbb{R}^{2n},\omega_0)$ whose closures are intersecting with coisotropic subspace $\mathbb{R}^{n,k}$ we construct relative versions of the Ekeland-Hofer capacities of the subsets with respect to $\mathbb{R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb{R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal{S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal{S}$.