Volume and Energy of Reeb Vector Fields

Abstract

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.