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

Martinazzi, Luca

doi:10.1016/j.anihpc.2012.12.007

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) no. 6, pp. 969-982

Voir la notice de l'article provenant de la source Numdam

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.

MR Zbl

DOI : 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 {\protect\emph{Q}-curvature} and large volume},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {969--982},
     publisher = {Elsevier},
     volume = {30},
     number = {6},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.12.007},
     mrnumber = {3132411},
     zbl = {1286.53018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.12.007/}
}

TY  - JOUR
AU  - Martinazzi, Luca
TI  - Conformal metrics on $ {\mathbb{R}}^{2m}$ with constant Q-curvature and large volume
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 969
EP  - 982
VL  - 30
IS  - 6
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.12.007/
DO  - 10.1016/j.anihpc.2012.12.007
LA  - en
ID  - AIHPC_2013__30_6_969_0
ER  -

%0 Journal Article
%A Martinazzi, Luca
%T Conformal metrics on $ {\mathbb{R}}^{2m}$ with constant Q-curvature and large volume
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 969-982
%V 30
%N 6
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.12.007/
%R 10.1016/j.anihpc.2012.12.007
%G en
%F 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) no. 6, pp. 969-982. doi: 10.1016/j.anihpc.2012.12.007

Cité par Sources :

Parcourir par

Geodesic

Parcourir par