Morse theory and Seiberg-Witten moduli spaces of 3-dimensional cobordisms, I

Motivated by a variant of Atiyah-Floer conjecture proposed in \cite{L2} and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) $Y$, perturbed by closed 2-forms of the form $r\ast d\text{ff}+w$, where $r\ge 1$, where $\text{ff}$ is a harmonic Morse function with certain linear growth at the ends of $Y$, and $w$ is a certain closed 2-form.