Pseudoholomorphic punctured spheres in R x (S^1 x S^2) : Properties and existence

TL;DR

This paper studies the structure of moduli spaces of pseudoholomorphic punctured spheres in a specific symplectic manifold, proving smoothness of components and conditions for boundary limits, advancing understanding in symplectic geometry.

Contribution

It establishes that all moduli space components are smooth manifolds and provides criteria for boundary limits of these spheres in R x (S^1 x S^2).

Findings

All moduli space components are smooth manifolds.

Necessary and sufficient conditions for boundary limits are given.

First detailed description of these moduli spaces in this setting.

Abstract

This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in R x (S^1 x S^2) 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 x S^2 to appear as the set of |s| --> infinity limits of the constant s in R slices of a pseudoholomorphic, multiply punctured sphere.