Integrability of scalar curvature and normal metric on conformally flat manifolds

Abstract

On a manifold (Rn,e2u|dx|2), we say u is normal if the Q-curvature equation that u satisfies (−Δ)n2u=Qgenu can be written as the integral form u(x)=1cn∫Rnlog⁡|y||x−y|Qg(y)enu(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.