This paper contains a characterization of Reeb vector fields of K-contact forms
in terms of J-holomorphic embeddings into the tangent unit sphere bundle. A
consequence of this characterization is that these vector fields are critical
points of a volume and an energy functionals defined on the set of unit vector
fields. Reeb vector fields on closed, K-contact Einstein manifolds are absolute
minimizers for the energy functional with a mean curvature correction. On
odd-dimensional Einstein manifolds of positive sectional curvature, these unit
vector fields are characterized by their minimizing property. It is also proved
that any closed flat contact manifold admits a parallelization by three critical
unit vector fields, one parallel (hence minimizing), the other two are Reeb
vector fields of contact forms, not Killing and not minimizers of any of the
volume or the energy functionals.