Seiberg-Witten Equations on Three-manifolds with Euclidean Ends

This paper comprises of the foundational work for some versions of Seiberg- Witten theory on 3-manifolds with Euclidean ends. A manifold with a Euclidean end, or an MEE for short, is a smooth, orientable 3-manifold formed by connect-summing a compact closed manifold with R3, whose metric is Euclidean outside a compact region (cf. Definition 2.2.2 below). We consider Seiberg-Witten equations ((2.2) below) on such manifolds, with a family of perturbations parametrized by t. See (2.4) below for the form of the perturbation 2-form. In contrast to the well-known theory on compact manifolds, which is essentially independent of the choice of metric or perturbation, the theory on non-compact manifolds typically depends crucially on the asymptotic conditions. Indeed, in our theory the cases of t = 0 and t \gt 0 behave quite differently.

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Published 1 January 2005