Strong closing property of contact forms and action selecting functors

TL;DR

The paper introduces a strong closing property for contact forms, provides a criterion based on monoidal functors, and conjectures its validity for standard contact forms on symplectic ellipsoid boundaries.

Contribution

It defines a new strong closing property for contact forms and establishes a criterion using monoidal functors, with a conjecture on its applicability to symplectic ellipsoid boundaries.

Findings

Proposed a sufficient criterion for strong closing property.

Formulated a conjecture for standard contact forms on symplectic ellipsoids.

Inspired by the $C^ ext{infinity}$ closing lemma in dimension three.

Abstract

We introduce a notion of strong closing property of contact forms, inspired by the $C^\infty$ closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.