Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

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].