Poincar\'e index and the volume functional of unit vector fields on punctured spheres

TL;DR

This paper establishes a lower bound for the volume of unit vector fields on punctured odd-dimensional spheres based on Poincaré indices, characterizes the minimizers, and explores multiple singularity configurations.

Contribution

It introduces a volume bound related to Poincaré indices on punctured spheres and identifies the vector fields that attain this minimum, extending understanding of singularities.

Findings

Lower bound for volume depending on Poincaré indices

Characterization of volume-minimizing vector fields

Discussion of multiple isolated singularities

Abstract

For $n\geq 1$, we exhibit a lower bound for the volume of a unit vector field on $\mathbb{S}^{2n+1}\backslash\{\pm p\}$ depending on the absolute values of its Poincar\'e indices around $\pm p$. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.