A note on prescribed Q-curvature

Mingxiang Li

Vol. 319 (2022), No. 1, 181–188

DOI: 10.2140/pjm.2022.319.181

Abstract

Let $(M,g)$ be a compact Riemannian manifold of dimension $5\le n\le 7$ with nonnegative scalar curvature and semipositive Q-curvature. Assume $(M,g)$ is not conformally equivalent to the standard sphere. If a prescribed positive smooth function $f$ is flat up to $n-4$ order at some maximum point, there exists a conformal metric $\tilde{g}=u^{4∕(n-4)}g$ such that $Q_{\tilde{g}}=f$. This is a natural higher order version of the result of Escobar and Schoen (1986).

Keywords

prescribed Q-curvature, Paneitz operator

Mathematical Subject Classification

Primary: 35J35, 53C21

Milestones

Received: 16 September 2021

Accepted: 4 June 2022

Published: 28 August 2022