Area minimizing unit vector fields on antipodally punctured unit 2-sphere and minimally immersed Klein Bottles

TL;DR

This paper investigates volume bounds for unit vector fields on antipodally punctured spheres and characterizes those that correspond to minimally immersed Klein bottles, revealing new geometric relationships.

Contribution

It establishes a lower volume bound depending on singularity indexes and links certain minimizing vector fields to Klein bottle immersions.

Findings

Derived a lower volume bound based on singularity indexes.

Identified conditions under which vector fields correspond to Klein bottle immersions.

Connected volume minimization with minimal Klein bottle immersions.

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. In addition, for minimizing vector fields having specific pair of indexes, we show that their image coincides with the image of minimally immersed Klein bottles.