#### Vol. 302, No. 1, 2019

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A compactness theorem on Branson's $Q$-curvature equation

### Gang Li

Vol. 302 (2019), No. 1, 119–179
##### Abstract

Let $\left(M,g\right)$ be a closed Riemannian manifold of dimension $5\le n\le 7$. Assume that $\left(M,g\right)$ is not conformally equivalent to the round sphere. If the scalar curvature ${R}_{g}$ is greater than or equal to 0 and the $Q$-curvature ${Q}_{g}$ is greater than or equal to 0 on $M$ with ${Q}_{g}\left(p\right)>0$ for some point $p\in M$, we prove that the set of metrics in the conformal class of $g$ with prescribed constant positive $Q$-curvature is compact in ${C}^{4,\alpha }$ for any $0<\alpha <1$.

##### Keywords
compactness, constant $Q$-curvature metrics, blowup argument, positive mass theorem, maximum principle, fourth order elliptic equations
##### Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 35B50, 35J61, 53A30