Compactness of Conformal Metrics with $L^p$-Bounded $Q$-Curvature on Closed Smooth Riemannian Manifolds

Let $(M^n,g)$ be a smooth closed Riemannian manifold of dimension $n\ge 5$ with positive Yamabe invariant and semi-positive $Q$-curvature. We establish a precompactness result in the $C^{\alpha }$-Hölder topologie on the space of Riemannian metrics, for some $\alpha >0$, for the set of metrics $\tilde{g}$ conformal to $g$, with volume equal to that of the standard sphere ${\mathbb{S}}^n$, whose $Q$-curvature is nonnegative and uniformly bounded in $L^p(M,\tilde{g})$ for some $p>\frac{n}{4}$, and whose first positive eigenvalue of the Laplace-Beltrami operator satisfies ${\lambda }_1(M,\tilde{g})\ge n+\frac{1}{\mathrm{\Lambda }}$ for some positive constant $\mathrm{\Lambda }$.