# Prescribing the $\bar Q^{\prime}$-Curvature on Pseudo-Einstein CR 3-Manifolds

## TL;DR

This paper investigates the problem of prescribing the $ar Q^{\

## Contribution

It introduces new methods for prescribing $ar Q^{\

## Key findings

- Prescribes positive CR pluriharmonic functions on compact pseudo-Einstein CR 3-manifolds.
- Establishes existence of solutions on the Heisenberg group under mild conditions.
- Identifies two types of solutions: normal with isoperimetric properties and non-normal with biharmonic terms.

## Abstract

In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality and non-normal ones that have a biharmonic leading term.