Remarks on GJMS operator of order six

Abstract

We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n\ge 10$, let $(M^n,g)$ 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 $f$.

to appear Pacific Journal of Mathematics

References in corpus (1)

• Explicit formulas for GJMS-operators and $Q$-curvatures

Cited by in corpus (1)

• Compactness of singular solutions to the sixth order GJMS equation