## Integrability of scalar curvature and normal metric on conformally flat manifolds

created by wang on 13 Nov 2020

[BibTeX]

Published Paper

Inserted: 13 nov 2020
Last Updated: 13 nov 2020

Journal: Journal of Differential Equations
Volume: 265
Pages: 1353-1370
Year: 2017

ArXiv: 1707.04361 PDF

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.