  We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends
(connected sums of $\R^3$ and a compact manifold) with perturbations which
approximate $*dx_3$ at infinity, and describe the structure of the moduli
spaces. The setup is inspired by Taubes's program of relating the 4-dimensional
Seiberg-Witten invariant with `singular Gromov invariants' and has related
applications.
