Integrability of scalar curvature and normal metric on conformally flat manifolds

TL;DR

This paper demonstrates that integrability conditions on scalar curvature ensure the metric's normality in conformally flat manifolds and establishes a bi-Lipschitz equivalence theorem for such metrics.

Contribution

It proves that integrability of negative scalar curvature parts guarantees the metric is normal and introduces a bi-Lipschitz equivalence result for conformally flat metrics.

Findings

Negative scalar curvature integrability implies normality.

Established bi-Lipschitz equivalence for conformally flat metrics.

Connected scalar curvature conditions to metric regularity.

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.