We define and study the harmonic curves on domains in \(\mathbb {R}^n\) into the first Heisenberg group \({\mathbb {H}}^{1}\) . These are the \(C^2\) -regular mappings which are critical points of the second Dirichlet energy and satisfy the weak isotropicity condition. We investigate the geometry of such curves including the comparison and maximum principles, the Harnack inequalities, the Liouville theorems, the existence results, the Phragmèn–Lindelöf theorem, as well as the three spheres theorem.
