We examine the space of
conformally compact metrics g on the interior of a compact manifold with boundary
which have the property that the kth elementary symmetric function of
the Schouten tensor Ag is constant. When k = 1 this is equivalent to the
familiar Yamabe problem, and the corresponding metrics are complete with
constant negative scalar curvature. We show for every k that the deformation
theory for this problem is unobstructed, so in particular the set of conformal
classes containing a solution of any one of these equations is open in the
space of all conformal classes. We then observe that the common intersection
of these solution spaces coincides with the space of conformally compact
Einstein metrics, and hence this space is a finite intersection of closed analytic
submanifolds.
