Monopoles and Landau-Ginzburg Models I

Abstract

The end point of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is any 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 any finite energy solution to the perturbed equations on $\mathbb{C}\times\Sigma$ is necessarily trivial. The second asserts that any small energy solutions on $\mathbb{H}^2_+\times\Sigma$ 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