  Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of
harmonic functions, i.e., functions that are critical points of the Dirichlet
energy, but no analogous theorem is known for $H$-minimal surfaces in the
three-dimensional Heisenberg group $\mathbb{H}$, which are known to have
singularities. In this paper, we introduce a definition of intrinsic Dirichlet
energy for surfaces in $\mathbb{H}$ and study the critical points of this
energy, which we call contact harmonic graphs. Nearly flat regions of
$H$-minimal surfaces can often be approximated by such graphs. We give a
calibration condition for an intrinsic Lipschitz graph to be energy-minimizing,
construct energy-minimizing graphs with a variety of singularities, and prove a
first variation formula for the energy of intrinsic Lipschitz graphs and
piecewise smooth intrinsic graphs.
