# Conformal metrics on $R_{2m}$ with constant $Q$-curvature

### Luca Martinazzi

Rutgers University, Piscataway, Switzerland

## Abstract

We study the conformal metrics on $R_{2m}$ with constant $Q$-curvature $Q∈R$ having finite volume, particularly in the case $Q≤0$. We show that when $Q<0$ such metrics exist in $R2m$ 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 $e_{2p}g_{R2m}$, where $p$ is a polynomial such that $2≤gp≤2m−2$ and $sup_{R2m}p<+∞$. In dimension $4$, such metrics correspond to the polynomials $p$ of degree $2$ with $lim_{∣x∣→+∞}p(x)=−∞$.

## Cite this article

Luca Martinazzi, Conformal metrics on $R_{2m}$ with constant $Q$-curvature. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 4, pp. 279–292

DOI 10.4171/RLM/525