A correspondence between genus one minimal Lawson surfaces of ${\mathbb{S}}^3(2)$ and area-minimizing unit vector fields on the antipodally punctured unit 2-sphere

A correspondence is established between a class of minimal immersed surfaces of ${\mathbb{S}}^3(2)$ and area-minimizing unit vector fields defined on the antipodally punctured unit sphere ${\mathbb{S}}^2\mathrm{\backslash }\{N,S\}$. As a consequence, we establish a stability relation for Lawson cylinders in ${\mathbb{S}}^3(2)$.