Removable Singularities of Magnetic Monopoles on Sasakian Manifolds

Abstract

<p>In this thesis I consider the problem of removable point singularities for static monopole fields defined on a three-dimensional compact Riemannian manifold Mˆ. A sucient condition for smooth extension of the associated hermitian vector bundle, the monopole connection and associated Higgs field is formulated in terms of an auxiliary system of equations, to be satisfied by the Higgs field. This ensures the availability of methods of complex analysis for the extension problem, when lifted to the Riemannian cone defined by Mˆ × R. A further requirement to be satisfied for this purpose is that Mˆ be Sasakian, at least strictly so in a neighbourhood of the singular point. This is shown to follow from the existence of a geodesible Killing vector field, on which the Ricci tensor of the Riemannian metric satisfies a natural positivity condition. For a full statement of the main Theorem, cf. Theorem 0.0.1 at the conclusion of the introduction.</p>