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

Martinazzi, Luca

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 969-982.

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

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