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We define and study the harmonic curves on domains in Rⁿ into the first Heisenberg group H¹. These are the C²-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 Phragm\`en-Lindel\"of theorem, as well as the three spheres theorem.
Adamowicz et al. (Mon,) studied this question.