Hyperbolic Multi-Monopoles With Arbitrary Mass

TL;DR

This paper proves the existence of hyperbolic monopoles with any given mass and magnetic charge, extending previous results limited to integral masses and charge one monopoles.

Contribution

It introduces a new method to construct monopoles with arbitrary mass on hyperbolic space using Taubes' gluing technique and weighted Sobolev spaces.

Findings

Existence of monopoles with arbitrary mass on hyperbolic 3-space.

Construction method based on gluing charge one monopoles.

Analysis ensuring absence of point spectra in the weighted Sobolev space.

Abstract

On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic charge and arbitrary mass. Previously, aside from charge one monopoles, existence was known only for monopoles with integral mass (since these arise from U(1) invariant instantons on Euclidean 4-space). The method of proof is based on Taubes' gluing procedure, using well-separated, explicit, charge one monopoles. The analysis is carried out in a weighted Sobolev space and necessitates eliminating the possibility of point spectra.