Semiclassical limit of Polyakov-Liouville measure and Q-Curvature Uniformization on even-dimensional manifolds

We study the semiclassical limit of the Polyakov-Liouville measure ${\mathit{\nu }}_{\gamma }$, which is a non-Gaussian measure on $H^{-\text{eps}}(M)$ that has recently been extended from Riemann surfaces to general Riemannian manifolds $(M,g)$ of even dimension. We show that under an appropriate rescaling in the semiclassical limit as $\gamma \to 0$, the normalized Polyakov-Liouville measure ${\text{Q}}_{\gamma }$ concentrates on the unique smooth weight $u$ for which the conformal metric $e^{2u}g$ on $M$ has constant $Q$-curvature.