Energy and topology of singular unit vector fields on S³

Abstract

The energy of unit vector fields on odd-dimensional spheres is a functional that has a minimum in dimension 3 and an infimum in higher dimensions. Vector fields with isolated singularities arise naturally in the study of this functional. We consider the class of fields in S 3 having two antipodal singularities. We prove a lower bound, attained for the radial vector field, for the energy of this class of fields in terms of the indices of the singularities. A similar inequality is not to be expected in other dimensions.

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

“…Additionally, as discussed in [6] for the energy functional, given a number (greater than two) of isolated singularities, it is possible to find a unit vector field having these singularities and with volume arbitrarily close to the volume of the radial vector field. This may be done by the following argument: put two singularities in antipodal points ±p and every remain singularity in a neighborhood near the south pole −p, for example.…”