Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles

TL;DR

This paper establishes perturbation methods for PU(2) monopole equations, ensuring transversality and compactness of the moduli space, which are crucial for understanding the geometric and topological properties of these monopoles.

Contribution

It introduces perturbations that achieve transversality and Uhlenbeck compactness for the PU(2) monopole moduli space, advancing the analytical foundation of the theory.

Findings

Existence of perturbations for PU(2) monopole equations

Achieved transversality in the moduli space

Established Uhlenbeck compactification

Abstract

This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality properties and an Uhlenbeck compactification for this perturbed moduli space.