Minimal Legendrian surfaces in the tangent sphere bundle of 𝕊³

Abstract

In this paper we study minimal Legendrian surfaces Σ \Sigma immersed in tangent sphere bundle T 1 S 3 T_1{\mathbb {S}}^3 . We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature.