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

Abstract

We study the conformal metrics on $\R^{2m}$ with constant Q-curvature $Q$ having finite volume, particularly in the case $Q\leq 0$. We show that when $Q<0$ such metrics exist in $\R^{2m}$ 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 $e^{2p}g_{\R^{2m}}$, where $p$ is a polynomial such that $2\leq \deg p\leq 2m-2$ and $\sup_{\R^{2m}}p<+\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with $\lim_{|x|\to+\infty}p(x)=-\infty$.