The tangential Cauchy-Riemann complex on the Heisenberg group Via Conformal Invariance

TL;DR

This paper leverages the conformal equivalence between the Heisenberg group and the CR sphere to solve the tangential Cauchy-Riemann operator on the Heisenberg group, advancing understanding in CR geometry.

Contribution

It introduces a method to transfer solutions of the tangential Cauchy-Riemann operator from the CR sphere to the Heisenberg group using conformal invariance.

Findings

Solved the tangential Cauchy-Riemann operator on the Heisenberg group

Established a link between CR sphere and Heisenberg group via conformal invariance

Enhanced techniques for analysis on CR manifolds

Abstract

The Heisenberg group $\mathbb{H}^1$ is known to be conformally equivalent to the CR sphere $\mathbb{S}^3$ minus a point. We use this fact, together with the knowledge of the tangential Cauchy-Riemann operator on the compact CR manifold $\mathbb{S}^3$, to solve the corresponding operator on $\mathbb{H}^1$.