A theorem about energy and volume of vector fields with the “proportional volume property”

In this paper, we define a certain proportional volume property for a unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum, and that 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 S2k+1.