For n ≥ 1, we exhibit a lower bound for the volume of a unit vector field on S 2n+1 \{±p} depending on the absolute values of its Poincaré indices around ±p. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.

No abstract

We present in this paper a "boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.

In this paper, we define a certain proportional volume property for an unit vector field on a spherical domain in S 3 . 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 S 2k+1 .

We present in this paper a "boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of three-dimensional Euclidean sphere.