Conformal metrics on ℝ^2m with constant <em>Q</em>-curvature

Abstract

We study the conformal metrics on $\R{2m}$ with constant $Q$-curvature $Q\in\R{}$ 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 correspond to the polynomials $p$ of degree $2$ with $\lim_{|x|\to+\infty}p(x)=-\infty$.