A Compactness Result for $\mathcal{H}-$holomorphic Curves in Symplectizations

TL;DR

This paper establishes a compactness result for a class of modified pseudoholomorphic curves called $\\mathcal{H}$-holomorphic curves in symplectizations, under certain bounds on harmonic forms, advancing understanding of their moduli spaces.

Contribution

It provides the first compactness theorem for $\\mathcal{H}$-holomorphic curves with harmonic form bounds, extending classical pseudoholomorphic curve theory.

Findings

Moduli space of $\\mathcal{H}$-holomorphic curves can be compactified under harmonic form bounds.

Introduces techniques to handle harmonic perturbations in symplectization settings.

Lays groundwork for further analysis of $\\mathcal{H}$-holomorphic curves in symplectic topology.

Abstract

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $\mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.