# A new proof of the $C^\infty$ regularity of $C^2$ conformal mappings on the Heisenberg group

## TL;DR

This paper presents a novel proof demonstrating that $C^2$ smooth conformal mappings on the Heisenberg group are infinitely differentiable, using hypoelliptic operators and quasiconformal flows instead of nonlinear potential theory.

## Contribution

It introduces a new proof technique for $C^
abla$ regularity of conformal maps on the Heisenberg group, avoiding nonlinear potential theory and leveraging hypoelliptic operators.

## Key findings

- Proof avoids nonlinear potential theory
- Relies on hypoellipticity of Hörmander operators
- Establishes $C^
abla$ regularity for conformal mappings

## Abstract

We give a new proof for the $C^\infty$ regularity of $C^2$ smooth conformal mappings of the sub-Riemannian Heisenberg group. Our proof avoids any use of nonlinear potential theory and relies only on hypoellipticity of H\"ormander operators and quasiconformal flows. This approach is inspired by prior work of Sarvas and Liu.