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

Iury Domingos ()
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Iury Domingos: Université de Lorraine, CNRS, IECL

Partial Differential Equations and Applications, 2022, vol. 3, issue 3, 1-20

Abstract: Abstract In this work, we consider the model of $${{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}$$ S 2 × R isometric to $${\mathbb {R}}^3{\setminus } \{0\}$$ R 3 \ { 0 } , endowed with a metric conformally equivalent to the Euclidean metric of $${\mathbb {R}}^3$$ 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 $${{\,\mathrm{{\mathbb {S}}^2\times {\mathbb {R}}}\,}}$$ S 2 × R with the same non-constant Gauss map differ by only two types of ambient isometries: either $$f=({{\,\mathrm{\mathrm {Id}}\,}},T)$$ f = ( Id , T ) , where T is a translation on $${\mathbb {R}}$$ R , or $$f=({\mathcal {A}},T)$$ f = ( A , T ) , where $${\mathcal {A}}$$ A denotes the antipodal map on $${\mathbb {S}}^2$$ S 2 . This means that any minimal immersion is determined by its conformal structure and its Gauss map, up to those isometries.

Keywords: Minimal surface; Gauss map; Conformal immersions; Homogenous 3-manifolds; Primary 53C42 Secondary 53A10; 53C30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00174-3

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