Calculus of Variations and Geometric Measure Theory
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L. Martinazzi

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

created by martinazz on 06 Feb 2009
modified on 17 Jul 2018

[BibTeX]

Published Paper

Inserted: 6 feb 2009
Last Updated: 17 jul 2018

Journal: Rend. Lincei Mat. Appl.
Volume: 19
Pages: 279-292
Year: 2008

ArXiv: 0805.0749 PDF

Abstract:

We study the conformal metrics on $\R^{2m}$ with constant Q-curvature $Q$ having finite volume, particularly in the case $Q\leq 0$. We show that when $Q<0$ such metrics exist in $\R^{2m}$ if and only if $m>1$. Moreover we study their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we treated in a recent paper. When Q=0, we show that such metrics have the form $e^{2p}g_{\R^{2m}}$, where $p$ is a polynomial such that $2\leq \deg p\leq 2m-2$ and $\sup_{\R^{2m}}p<+\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with $\lim_{
x
\to+\infty}p(x)=-\infty$.

Keywords: Q-curvature, Concentration-compactness, Conformal geometry


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