This monograph is a continuation of [C. H. Taubes, ‘\(L^2\)-moduli spaces on 4-manifolds with cylindrical ends’, Monographs in Geometry and Topology. 1 (1993)]reviewed above. It is divided into three parts. The first deals with \(L^2\)-moduli spaces on a manifold with cylindrical ends providing a local description of them when the energy is fixed. The second introduces a local thicking of the \(L^2\)-moduli spaces and proves various local structure theorems for it. The third discusses the example of a four-manifold with an end modeled on a circle bundle over a Riemann surface. As an illustration of the proposed techniques a vanishing theorem of the Donaldson invariants which contain an embedded nontrivial circle bundle over a torus is proved.