Invariant surfaces of the heisenberg groups

We fix a lefl-invariant metric g in the Heisenberg group, H₃, and give a complete classification of the constant mean curvature surfaces (including minimal) which are invariant with respect to 1-dimensional closed subgroups of the connected component of the isometry group of (H₃, g). In addition to finding new examples, we organize in a common framework results that have appeared in various forms in the literature, by the systematic use of Riemannian transformation groups. Using the existence of a family of spherical surfaces for all values of nonzero mean curvature, we show that there are no complete graphs of constant mean curvature. We extend some of these results to the higher dimensional Heisenberg groups H_2n+1.