Area Minimizing Unit Vector Fields on Antipodally Punctured Unit 2-Sphere

TL;DR

This paper establishes a lower bound for the volume of unit vector fields on antipodally punctured spheres, linking the bound to the singularity indexes and geometric properties of the vector field.

Contribution

It introduces a new lower volume bound for tangent unit vector fields on antipodally punctured spheres based on singularity indexes and ellipse geometry.

Findings

Derived a lower volume bound depending on singularity indexes.

Connected the volume bound to the geometric shape of an ellipse.

Enhanced understanding of vector field constraints on punctured spheres.

Abstract

We provide a lower value for the volume of a unit vector field tangent to an antipodally Euclidean sphere $\mathbb{S}^2$ depending on the length of an ellipse determined by the indexes of its singularities.