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.
|Original language||English (US)|
|Number of pages||48|
|Journal||Annali della Scuola Normale Superiore di Pisa - Classe di Scienze|
|State||Published - 2023|
ASJC Scopus subject areas
- Theoretical Computer Science
- Mathematics (miscellaneous)