Second variation of volume and energy of vector fields. Stability of Hopf vector fields

In this paper we compute the Hessian of the volume of unit vector fields at a minimal one. We also find the Hessians of a family of functionals thus generalizing the known results concerning second variation of the energy or total bending. We use them to study the stability of Hopf vector fields on and to show that they are stable for , but that for there is such that for the index is at least . Received May 10, 1999 / Published online April 12, 2001