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

Abstract

We study conformal metrics $g_u=e^{2u}|dx{|}^2$ on ${\mathbb{R}}^{2m}$ with constant $Q$-curvature $Q_{g_u}\equiv (2m-1)!$ (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 $V\in [V^{⁎},\infty )$ there is a conformal metric $g_u=e^{2u}|dx{|}^2$ on ${\mathbb{R}}^6$ with $Q_{g_u}\equiv 5!$ and $\mathrm{vol}(g_u)=V$. 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 $V_m>\mathrm{vol}(S^{2m})$ such that for every $V\in (0,V_m]$ there is a conformal metric $g_u=e^{2u}|dx{|}^2$ on ${\mathbb{R}}^{2m}$ with $Q_{g_u}\equiv (2m-1)!$, $\mathrm{vol}(g)=V$. 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.