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

Gang Li

Vol. 302 (2019), No. 1, 119–179

DOI: 10.2140/pjm.2019.302.119

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

Milestones

Received: 21 December 2017

Revised: 4 January 2019

Accepted: 13 March 2019

Published: 5 November 2019