Integrability of scalar curvature and normal metric on conformally flat manifolds

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.