  We use variational arguments to introduce a notion of mean curvature for
surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\'eodory
distance. By analyzing the first variation of area, we characterize C^2
stationary surfaces for the area as those with mean curvature zero (or constant
if a volume-preserving condition is assumed) and such that the characteristic
curves meet orthogonally the singular curves. Moreover, a Minkowski type
formula relating the area, the mean curvature, and the volume is obtained for
volume-preserving area-stationary surfaces enclosing a given region.
  As a consequence of the characterization of area-stationary surfaces, we
refine previous Bernstein type theorems in order to describe entire
area-stationary graphs over the xy-plane in H^1. A calibration argument shows
that these graphs are globally area-minimizing.
  Finally, by using the known description of the singular set, the
characterization of area-stationary surfaces, and the ruling property of
constant mean curvature surfaces, we prove our main results where we classify
volume-preserving area-stationary surfaces in H^1 with non-empty singular set.
In particular, we deduce the following counterpart to Alexandrov uniqueness
theorem in Euclidean space: any compact, connected, C^2 surface in H^1
area-stationary under a volume constraint must be congruent with a rotationally
symmetric sphere obtained as the union of all the geodesics of the same
curvature joining two points. As a consequence, we solve the isoperimetric
problem in H^1 assuming C^2 smoothness of the solutions.
