  In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of
properly embedded minimal annuli foliated by circles. In this paper we show
that this family contains all properly embedded minimal annuli. We use the
description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic
harmonic maps $G : \mathbb{C} \to \mathbb{S}^2$ of finite type. Due to the
algebraic geometric correspondence of Hitchin [14], these harmonic maps are
parametrized by hyperelliptic algebraic curves together with Abelian
differentials with prescribed poles. We deform annuli by deforming spectral
data in the corresponding moduli space. Along this deformation we control the
flux and we preserve embeddedness. The center of the theory concerns the study
of singularities of the flow. In particular we open and close nodes of singular
spectral curves. This approach applies also to mean convex Alexandrov embedded
cmc annuli in $\mathbb{S}^3$ [12].

Differential Geometry