On the higher order conformal covariant operators on the sphere

Abstract

We will show that in the conformal class of the standard metric $g_{S^n}$ on $S^n$, the scaling invariant functional $({\mu }_g(S^n){)}^{\frac{2m-n}{n}}{\int }_{S^n}{Q}_{2m,g}d{\mu }_g$ maximizes at $g_{S^n}$ when $n$ is odd and $m=\frac{n+1}{2}$ or $\frac{n+3}{2}$. For $n$ odd and $m\ge \frac{n+5}{2}$, $g_{S^n}$ is not stable and the functional has no local maximizer. Here $Q_{2m,g}$ is the $2m$th order $Q$-curvature.

19 pages