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

### Luca Martinazzi

Rutgers University, United States

## Abstract

We study conformal metrics $g_{u}=e_{2u}∣dx∣_{2}$ on $R_{2m}$ with constant $Q$-curvature $Q_{g_{u}}≡(2m−1)!$ (notice that $(2m−1)!$ is the $Q$-curvature of $S_{2m}$) and finite volume. When $m=3$ we show that there exists $V_{⁎}$ such that for any $V∈[V_{⁎},∞)$ there is a conformal metric $g_{u}=e_{2u}∣dx∣_{2}$ on $R_{6}$ with $Q_{g_{u}}≡5!$ and $vol(g_{u})=V$. This is in sharp contrast with the four-dimensional case, treated by C.-S. Lin. We also prove that when $m$ is odd and greater than $1$, there is a constant $V_{m}>vol(S_{2m})$ such that for every $V∈(0,V_{m}]$ there is a conformal metric $g_{u}=e_{2u}∣dx∣_{2}$ on $R_{2m}$ with $Q_{g_{u}}≡(2m−1)!$, $vol(g)=V$. This extends a result of A. Chang and W.-X. Chen. When $m$ is even we prove a similar result for conformal metrics of *negative $Q$*-curvature.

## Cite this article

Luca Martinazzi, Conformal metrics on $R_{2m}$ with constant $Q$-curvature and large volume. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 6, pp. 969–982

DOI 10.1016/J.ANIHPC.2012.12.007