The Paneitz-Sobolev constant of a closed Riemannian manifold and an application to the nonlocal $\mathbf{Q}$-curvature flow

In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $\geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)