Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S^{2n+1} is the Reeb vector field of a natural Sasakian structure on S^{2n+1}. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k,µ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study the stability (and instability) of the Reeb vector field ξ of a compact H-contact three-manifold with respect to the energy (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature . We study the stability of ξ considering separately: (1) the Sasakian case (Theorem 3.1 and Proposition 5.1); (2) the case where M is a generalized (k,µ)-space (Theorem 3.2), in such case we find examples of non-Killing stable harmonic vector fields; (3) the case where M is non-Sasakian H-contact with ξ minimal (Theorem 4.1 and Theorem 5.1). In particular, on the flat contact metric three-torus, energy and volume obtain the minimum but the Reeb vector field is energy and volume unstable (for a Riemannian 2-torus all harmonic unit vector fields are energy stable ). In the last section, we extend for the Reeb vector field of a compact K-contact manifold (Theorem 6.1 and Theorem 6.2) the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.