A theorem about vector fields with the "proportional volume property"

TL;DR

This paper introduces a proportional volume property for unit vector fields on spherical domains, proves the existence of a minimum volume achieved by Hopf vector fields, and explores related energy minimization in Euclidean spheres.

Contribution

It establishes a new volume property for vector fields on spheres and proves that Hopf vector fields minimize volume under this property, extending understanding of vector field optimization.

Findings

Volume of vector fields with the property has an absolute minimum equal to Hopf vector fields.

Examples of vector fields satisfying the property are provided.

Minimum energy of solenoidal vector fields coincides with Hopf flows on spherical boundaries.

Abstract

In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the minimum energy of solenoidal vector fields which coincides with a Hopf flow along the boundary of a spherical domain of an odd-dimensional euclidean sphere.