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

We construct contact forms with constant $Q^{\mathrm{\prime }}$-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 $P^{\mathrm{\prime }}$-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 $\sqrt{P^{\mathrm{\prime }}}$.