  We investigate the minimal surface problem in the three dimensional
Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric.
Using a particular surface measure, we characterize minimal surfaces in terms
of a sub-elliptic partial differential equation and prove an existence result
for the Plateau problem in this setting. Further, we provide a link between our
minimal surfaces and Riemannian constant mean curvature surfaces in H equipped
with different Riemannian metrics approximating the Carnot-Caratheodory metric.
We generate a large library of examples of minimal surfaces and use these to
show that the solution to the Dirichlet problem need not be unique. Moreover,
we show that the minimal surfaces we construct are in fact X-minimal surfaces
in the sense of Garofalo and Nhieu.
