Pseudoholomorphic punctured spheres in $\mathbb{R}{\times}(S^{1}{\times}S^{2})$: Properties and existence

Abstract

This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in $ℝ\times \left(S^1\times S^2\right)$ as defined by a certain natural pair of almost complex structure and symplectic form. This article proves that all moduli space components are smooth manifolds. Necessary and sufficient conditions are also given for a collection of closed curves in $S^1\times S^2$ to appear as the set of $\left|s\right|\to \infty$ limits of the constant $s\in ℝ$ slices of a pseudoholomorphic, multiply punctured sphere.