Harmonic intrinsic graphs in the Heisenberg group

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1367-1414
Number of pages48
JournalAnnali della Scuola Normale Superiore di Pisa - Classe di Scienze
Issue number3
StatePublished - 2023

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)


Dive into the research topics of 'Harmonic intrinsic graphs in the Heisenberg group'. Together they form a unique fingerprint.

Cite this