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

TL;DR

This paper proves a compactness result for metrics with prescribed constant positive $Q$-curvature on 5-dimensional manifolds under certain scalar curvature conditions, with additional estimates in higher dimensions.

Contribution

It establishes a new compactness theorem for the $Q$-curvature equation on 5-dimensional manifolds not conformally equivalent to the sphere, extending understanding of conformal geometry.

Findings

Set of conformal metrics with prescribed constant positive $Q$-curvature is compact in $C^{4, eta}$ topology.

Provides estimates for the $Q$-curvature equation in dimensions 6 and 7.

Shows non-compactness does not occur under specified curvature conditions.

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 $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ 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 $7$.