A gluing construction for periodic monopoles

In [4] and [6] Cherkis and Kapustin introduced the study of periodic monopoles (with singularities), that is, monopoles on R2 × S1 possibly singular at a finite collection of points. Four-dimensional moduli spaces of periodic monopoles with singularities are expected to provide examples of gravitational instantons, that is, complete hyperkähler four-manifolds with finite energy. In a previous paper [9] we proved that the moduli space of charge k periodic monopoles with n singularities is either empty or generically a smooth hyperkähler manifold of dimension 4(k−1). In this paper we settle the existence question, constructing periodic monopoles (with singularities) by gluing methods.