  The endpoint of this series of papers is to construct the monopole Floer
homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold
with toroidal boundary and $\omega$ 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
$\mathbb{C}\times\Sigma$ or $\mathbb{H}^2_+\times\Sigma$, where $\Sigma$ is any
compact Riemann surface of genus $\geq 1$. Our first result states that finite
energy solutions to the perturbed equations on $\mathbb{C}\times\Sigma$ are
necessarily trivial. The second states that small energy solutions on
$\mathbb{H}^2_+\times\Sigma$ necessarily have energy decaying exponentially in
the spatial direction. These results will lead eventually to the compactness
theorem in the second paper.

Geometric Topology;Differential Geometry