Poincaré index and the volume functional of unit vector fields on punctured spheres

Abstract

For $n\ge 1$, 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 Poincaré indices around $\pm p$. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.

Revised version. 15 pages, 2 figures