The energy of unit vector fields on the 3-sphere

The stability of the three-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of the three-dimensional Hopf vector field is compared and contrasted with that of the closely related Hopf map. Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere.