Local behaviour of sequences of three-dimensional generalised monopoles

Abstract

The purpose of this paper is to study the behaviour of sequences of generalised monopoles with a uniform bound on a certain L2-norm. We focus on the case that the target hyperKahler manifolds are Swann bundles. In 3-dimensional case, suppose that there exists an open submanifold Y' such that the hyperKahler potential along the monopoles has a uniform lower bound over Y'. Then we show that there exist convergent subsequences of generalised monopoles over any compact subset of Y'. Under similar assumptions, the same conclusion holds for the generalised harmonic spinors in dimension four.