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

Abstract

We study the conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature $Q$ 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 $2\le °p\le 2m-2$ and ${\sup }_{{\mathbb{R}}^{2m}}p<+\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with ${\lim }_{|x|\to +\infty }p(x)=-\infty$.

13 pages