Extremal metrics for the $Q^\prime$-curvature in three dimensions

TL;DR

This paper constructs special contact forms with constant $Q'$-curvature on certain three-dimensional CR manifolds, using variational methods and analysis of pseudodifferential operators.

Contribution

It introduces a method to find contact forms with constant $Q'$-curvature by minimizing a CR functional, and establishes the ellipticity of the $P'$-operator.

Findings

Construction of contact forms with constant $Q'$-curvature.

Proof that the $P'$-operator is elliptic pseudodifferential.

Asymptotic expansion of the Green's function for $ oot P'$.

Abstract

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.