In this paper we obtain criteria of stability for η-Einstein K-contact manifolds, for Sasakian manifolds of constant φ-sectional curvature and for 3-dimensional Sasakian manifolds. Moreover, we show that a stable compact Einstein contact metric manifold M is Sasakian if and only if the Reeb vector field ξ minimises the energy functional. In particular, the Reeb vector field of a Sasakian manifold M of constant φ-holomorphic sectional curvature +1 minimises the energy functional if and only if M is not simply connected.
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