Unit vector fields of minimum energy on quotients of spheres and stability of the Reeb vector field

Abstract

In this paper we show that, in contrast with the situation for the sphere S2n+1(κ), on a quotient S2n+1(κ)/Γ, Γ≠{Id}, the unit Hopf vector fields are the unique unit vector fields which minimize the energy. Moreover, we show that such vector fields are stable critical points with respect to the volume functional. Then, we study the stability question, with respect to energy and volume, of the Reeb vector field for H-contact, K-contact manifolds, Sasakian space forms and 3-Sasakian manifolds.