Remarks on GJMS operator of order six

Xuezhang Chen and Fei Hou

Vol. 289 (2017), No. 1, 35–70

DOI: 10.2140/pjm.2017.289.35

Abstract

We study analysis aspects of the sixth-order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, we present the expansions of Green’s function of $P_g^6$ with pole at this point. As a starting point of the study of $P_g^6$, we manage to give some existence results of the prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n\ge 10$, let $\left(M^n,g\right)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $\tilde{g}$ of $g$ such that its $Q$-curvature $Q_{\tilde{g}}^6$ equals $f$.

Keywords

sixth-order GJMS operator, prescribed $Q$-curvature problem, Green's function, mountain pass critical points

Mathematical Subject Classification 2010

Primary: 53A30, 53C21, 58J05

Secondary: 35B50, 35J08, 35J35

Milestones

Received: 6 March 2016

Revised: 11 October 2016

Accepted: 11 January 2017

Published: 12 May 2017