The end point of this series of papers is to construct the monopole Floer
homology for any pair (Y,ω), where Y is any compact oriented
3-manifold with toroidal boundary and ω is a suitable closed 2-form. In
the first paper, we exploit the framework of the gauged Landau-Ginzburg models
to address two model problems for the (perturbed) Seiberg-Witten moduli spaces
on either C×Σ or H+2​×Σ, where
Σ is any compact Riemann surface of genus ≥1. Our first result
states that any finite energy solution to the perturbed equations on
C×Σ is necessarily trivial. The second asserts that any
small energy solutions on H+2​×Σ necessarily have energy
decay exponentially in the spatial direction. These results will lead
eventually to the compactness theorem in the second paper.Comment: 56 pages. v2. We add an appendix explaining the case for higher genus
surfaces. v3. The statement of the main results are revised to incorporate
higher genus surfaces as wel