Integrability of scalar curvature and normal metric on conformally flat manifolds

On a manifold (Rⁿ ,e^2u | dx|²), we say u is normal if the Q-curvature equation that u satisfies (- Δ) ^n/2 u =Q_ge^nu can be written as the integral form u (x) = 1/c_n ∫_Rⁿ log ⁡ |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.