Monopoles over 4-manifolds containing long necks. I (Q2388859)

Many mathematicians attempted to express the Seiberg-Witten invariants of a closed \(\text{spin}^c\) 4-manifold \(Z\) cut along a hypersurface \(Y\), say \(Z = Z_1 \cup_Y Z_2\), in terms of relative invariants of the two pieces \(Z_1\) and \(Z_2\), using a 1-parameter family of metrics on \(Z\) by stretching along \(Y\) to get a long neck \([-T, T] \times Y\). The author of the present paper studies moduli spaces of Seiberg-Witten monopoles over \(\text{spin}^c\) Riemannian 4-manifolds with long necks. He perturbs the monopole equations by a closed 2-form on \(Y\). He discusses compactness of moduli spaces, exponential decay of monopoles, non-degeneracy of critical points and regularity of moduli spaces. As applications he proves two vanishing theorems for Seiberg-Witten invariants. Also he gives two appendices: how to patch together sequences of local gauge transformations, and a quantitative inverse function theorem.