A compactness theorem on Branson’s Q-curvature equation

Let $(M, g)$ be a closed Riemannian manifold of dimension $5 \leq n \leq 7$ . Assume that $(M, g)$ 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 $Q_{g} (p) > 0$ 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 $C^{4, α}$ for any $0 < α < 1$ .

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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

Authors

Gang Li
Beijing International Center for Mathematical Research Peking University Beijing China	Department of Mathematics Shandong University Ji’an China

Vol. 302, No. 1, 2019

Gang Li

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors