Conformal metrics on $R^{2m}$ with constant $Q$-curvature

Abstract

We study the conformal metrics on ${\mathbb{R}}^{2m}$ with constant $Q$-curvature $Q\in \mathbb{R}$ having finite volume, particularly in the case $Q\le 0$. We show that when $Q<0$ such metrics exist in $\mathbb{R}2m$ if and only if $m>1$. Moreover we study their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we treated in a recent paper. When $Q=0$, we show that such metrics have the form $e^{2p}{g}_{\mathbb{R}2m}$, where $p$ is a polynomial such that $2\le \mathrm{deg}p\le 2m-2$ and ${\sup }_{\mathbb{R}2m}p<+\infty$. In dimension $4$, such metrics correspond to the polynomials $p$ of degree $2$ with ${\lim }_{|x|\to +\infty }p(x)=-\infty$.