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.