  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.
