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

Abstract

In this work, we consider the model of \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) 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 Euclidean 3-space. We show as a main result that any two minimal conformal immersions in \({{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}\) with the same non-constant Gauss map differ by only two types of ambient isometries: either \(f=({{\,\mathrm{\mathrm {Id}}\,}},T)\), where T is a translation on \({\mathbb {R}}\), or \(f=({\mathcal {A}},T)\), where \({\mathcal {A}}\) denotes the antipodal map on \({\mathbb {S}}^2\). This means that any minimal immersion is determined by its conformal structure and its Gauss map, up to those isometries.