Conformal metrics on ${ℝ}^{2m}$ with constant Q-curvature and large volume
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, p. 969-982

We study conformal metrics ${g}_{u}={e}^{2u}{|dx|}^{2}$ on ${ℝ}^{2m}$ with constant Q-curvature ${Q}_{{g}_{u}}\equiv \left(2m-1\right)!$ (notice that $\left(2m-1\right)!$ is the Q-curvature of ${S}^{2m}$) and finite volume. When $m=3$ we show that there exists ${V}^{⁎}$ such that for any $V\in \left[{V}^{⁎},\infty \right)$ there is a conformal metric ${g}_{u}={e}^{2u}{|dx|}^{2}$ 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 $V\in \left(0,{V}_{m}\right]$ there is a conformal metric ${g}_{u}={e}^{2u}{|dx|}^{2}$ on ${ℝ}^{2m}$ with ${Q}_{{g}_{u}}\equiv \left(2m-1\right)!$, $\mathrm{vol}\left(g\right)=V$. 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.

DOI : https://doi.org/10.1016/j.anihpc.2012.12.007
Keywords: Q-curvature, Paneitz operators, GMJS operators, Conformal geometry
@article{AIHPC_2013__30_6_969_0,
author = {Martinazzi, Luca},
title = {Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature and large volume},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {6},
year = {2013},
pages = {969-982},
doi = {10.1016/j.anihpc.2012.12.007},
zbl = {1286.53018},
mrnumber = {3132411},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_6_969_0}
}
Martinazzi, Luca. Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature and large volume. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 6, pp. 969-982. doi : 10.1016/j.anihpc.2012.12.007. http://www.numdam.org/item/AIHPC_2013__30_6_969_0/

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