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

Abstract: We study the semiclassical limit of the Polyakov-Liouville measure $\boldsymbolν<em>γ$, which is a non-Gaussian measure on $H<sup>{-\eps}(M)$ 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 $γ\to0$, the normalized Polyakov-Liouville measure $\Q</em>γ$ concentrates on the unique smooth weight $u$ for which the conformal metric $e<sup>{2u}g$ on $M$ has constant $Q$-curvature.