Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Abstract

We provide a lower value for the volume of a unit vector field tangent to an antipodally punctured Euclidean sphere $\mathbb{S}^2$ depending on the length of an ellipse determined by the indexes of its singularities. We also exhibit minimizing vector fields $\vec{v}_k$ within each index class and show that they are the only ones that are sharp for the volume. These fields have areas given essentially by the length of ellipses depending just on the indexes in $N$ and $S$.