  This work generalizes a construction by Habermann and Jost of a canonical
metric in a Yamabe-positive conformal class, which uses the Green function of
the conformal Laplacian. In dimension $n=2k+1$, $2k+2$, or $2k+3$, if the
$k$-th GJMS operator $P_k$ admits a Green function, the constant term of its
singularity is shown to be a conformal density of weight $2k-n$, when
restricted to appropriate choices of conformal factor. When it is positive, it
is used to build a canonical metric in the conformal class. In the case of the
Paneitz-Branson operator $P_2$, in dimension 5, 6 or 7, we show a positiveness
result. In additition, we relate it to an asymptotic invariant of the manifold
obtained by stereographic projection via the Green function.
