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

Abstract

Let $(M,g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M,g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\ge 0$ and the $Q$-curvature $Q_g\ge 0$ on $M$ with $Q_g(p)>0$ 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 $0<\alpha <1$. We also give some estimates for dimension $6$ and $7$.

Comments are welcome

References in corpus (1)

• The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight

Cited by in corpus (1)

• Uniqueness of conformal metrics with constant Q-curvature on closed Einstein manifolds