Harmonic intrinsic graphs in the Heisenberg group

Minimal surfaces in Rⁿ 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 H, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in 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.