*Published Paper*

**Inserted:** 13 nov 2020

**Last Updated:** 13 nov 2020

**Journal:** Journal of Differential Equations

**Volume:** 265

**Pages:** 1353-1370

**Year:** 2017

**Abstract:**

On a manifold $(\mathbb{R}^n, e^{2u}

dx

^2)$, we say $u$ is normal if the
$Q$-curvature equation that $u$ satisfies $(-\Delta)^{\frac{n}{2}} u = Q_g
e^{nu}$ can be written as the integral form $u(x)=\frac{1}{c_n}\int_{\mathbb
R^n}\log\frac{

y

}{

x-y

}Q_g(y)e^{nu(y)}dy+C$. In this paper, we show that the
integrability assumption on the negative part of the scalar curvature implies
the metric is normal. As an application, we prove a bi-Lipschitz equivalence
theorem for conformally flat metrics.