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.
Categories: math.DG math.AP