Ricci-positive geodesic flows and point-completion of static monopole fields

Abstract

Let (Mˆ, g) be a compact, oriented Riemannian three-manifold corresponding to the metric point-completion M ∪{P₀} of a manifold M, and let ξ denote a geodesible Killing unit vector field on Mˆ such that the Ricci curvature function Ricg (ξ ) > 0 everywhere, and is constant outside a compact subset K ⊂⊂ M. Suppose further that (E, ∇, ϕ) supply the essential data of a monopole field on M, smooth outside isolated singularities all contained in K. The main theorem of this article provides a sufficient condition for smooth extension of (E, ∇, ϕ) across P₀, in terms of the Higgs potential Φ, defined in a punctured neighbourhood of P₀ by ∇ξΦ − 2i[ϕ, Φ] = ϕ . The sufficiency condition is expressed by a system of equations on the same neighbourhood, which can be effectively simplified in the case that Mˆ is a regular Sasaki manifold, such as the round S³.