Harmonic intrinsic graphs in the Heisenberg group

TL;DR

This paper introduces a new notion of intrinsic Dirichlet energy for surfaces in the Heisenberg group and studies contact harmonic graphs as analogs of harmonic functions for minimal surfaces with singularities.

Contribution

It defines contact harmonic graphs in the Heisenberg group, establishes calibration conditions, constructs energy-minimizing graphs with singularities, and derives a first variation formula.

Findings

Contact harmonic graphs can approximate flat regions of $H$-minimal surfaces.

Calibration conditions identify energy-minimizing intrinsic Lipschitz graphs.

Constructed examples include graphs with various singularities.

Abstract

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.