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

TL;DR

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

Contribution

It introduces a variational approach to find contact forms with constant $Q'$-curvature and analyzes the $P'$-operator as an elliptic pseudodifferential operator in CR geometry.

Findings

Constructed contact forms with constant $Q'$-curvature.

Established the ellipticity of the $P'$-operator.

Computed 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 $\sqrt{P^\prime}$.