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

Martinazzi, Luca

Conformal metrics on

ℝ^{2 m}

with constant Q-curvature and large volume

Martinazzi, Luca

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 969-982 / Harvested from Numdam

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Résumé

We study conformal metrics $g_{u} = e^{2 u} {| d x |}^{2}$ on $ℝ^{2 m}$ with constant Q-curvature $Q_{g_{u}} \equiv (2 m - 1)!$ (notice that $(2 m - 1)!$ is the Q-curvature of $S^{2 m}$ ) 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 $g_{u} = e^{2 u} {| d x |}^{2}$ on $ℝ^{6}$ with $Q_{g_{u}} \equiv 5!$ 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 $V_{m} > vol (S^{2 m})$ such that for every $V \in (0, V_{m}]$ there is a conformal metric $g_{u} = e^{2 u} {| d x |}^{2}$ on $ℝ^{2 m}$ with $Q_{g_{u}} \equiv (2 m - 1)!$ , $vol (g) = V$ . 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.

Publié le : 2013-01-01

MR 3132411 | Zbl 1286.53018

DOI : https://doi.org/10.1016/j.anihpc.2012.12.007

@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},
     volume = {30},
     year = {2013},
     pages = {969-982},
     doi = {10.1016/j.anihpc.2012.12.007},
     mrnumber = {3132411},
     zbl = {1286.53018},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 30 (2013) pp. 969-982. doi : 10.1016/j.anihpc.2012.12.007. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_6_969_0/

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