*preprint*

**Inserted:** 17 jul 2018

**Last Updated:** 17 jul 2018

**Year:** 2012

**Abstract:**

We study conformal metrics on R^{{2m}} with constant Q-curvature and finite
volume. When m=3 we show that there exists V** such that for any V\in
V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature of
S^6, and vol(g)=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\in (0,V_m there is
a conformal metric g on R ^{{2m}} with Q_{g} = Q-curvature of S^{{2m},} 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.
**