Harmonic Curves From Euclidean Domains to Heisenberg Group H1

TL;DR

This paper introduces and analyzes harmonic curves from Euclidean domains into the Heisenberg group, exploring their geometric properties and establishing fundamental principles and theorems related to these mappings.

Contribution

It defines harmonic curves into the Heisenberg group and investigates their geometric and analytical properties, including maximum principles and existence results.

Findings

Established comparison and maximum principles for harmonic curves.

Proved Harnack inequalities and Liouville theorems for these curves.

Derived Phragmèn-Lindelöf and three spheres theorems in this context.

Abstract

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 Phragm\`en-Lindel\"of theorem, as well as the three spheres theorem.