preprint

Inserted: 17 jul 2018

Year: 2013

Abstract:

We study conformal metrics on $R^3$, i.e., metrics of the form $g_u=e^{2u}
dx
^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation $$ (-\Delta)^\frac32 u = 2 e^{3u}$$ in $R^3$ $$V:=\int_{\mathbb{R}^{3}e{3u}dx<\infty,$$} where $V$ is the volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $V\le 2\pi^2=
S^3
$. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.