The Gauss map of minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$

In this work, we consider the model of $\mathbb{S}^2\times\mathbb{R}$ isometric to $\mathbb{R}^3\setminus \{0\}$, endowed with a metric conformally equivalent to the Euclidean metric of $\mathbb{R}^3$, and we define a Gauss map for surfaces in this model likewise in the $3-$Euclidean space. We show as a main result that any two minimal conformal immersions in $\mathbb{S}^2\times\mathbb{R}$ with the same non-constant Gauss map differ by only two types of ambient isometries: either $f=(\mathrm{id},T)$, where $T$ is a translation on $\mathbb{R}$, or $f=(\mathcal{A},T)$, where $\mathcal{A}$ denotes the antipodal map on $\mathbb{S}^2$. Moreover, if the Gauss map is singular, we show that it is necessarily constant, and then only vertical cylinders over geodesics of $\mathbb{S}^2$ in $\mathbb{S}^2\times\mathbb{R}$ appear with this assumption. We also study some particular cases, among them we prove that there is no minimal conformal immersion in $\mathbb{S}^2\times\mathbb{R}$ which the Gauss map is a non-constant anti-holomorphic map.