Existence of complete conformal metrics on ${\mathbb{R}}^n$ with prescribed Q-curvature

Abstract

Given a smooth function $f(x)$ on ${\mathbb{R}}^n$ which is positive somewhere and satisfies $f(x)=O(|x{|}^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx{|}^2$ with finite total Q-curvature such that its Q-curvature equals to $f(x)$.

14 pages