This Chapter occupies a central position in the book as a whole. Building on the results of Chapter 3, we develop the theory of instantons over non-compact manifold with tubular ends. The theory can be considered as a modification of the standard set-up over compact manifolds and we shall see that many of the results go over to the non-compact case; in particular the instanton equations become non-linear Fredholm equations and we get finite-dimensional moduli spaces of solutions. The original reference for most of this material is [45] (in the more general setting of manifolds with ‘periodic ends’).

The work in this Chapter falls into three main parts. First we study the decay of instantons over tubular ends. We shall see that, under suitable conditions, an instanton with L2 curvature can be represented by a connection form which decays exponentially down the tube. This can be regarded as a counterpart of the elliptic regularity theory for the instanton equations; it implies that all of the possible natural definitions of ‘decaying instantons’ are equivalent. The proofs are straightforward modifications of those of previous results for the case when the cross-section is a 3-sphere. In the second part we set up function spaces to present the instanton equations as non-linear Fredholm equations. We shall consider two cases, depending up on the form of the limiting connections over the cross-sections of the tubes.