  We study existence and uniqueness results for the Yamabe problem on
non-compact manifolds of negative curvature type. Our first existence and
uniqueness result concerns those such manifolds which are asymptotically
locally hyperbolic. In this context, our result requires only a partial $C^2$
decay of the metric, namely the full decay of the metric in $C^1$ and the decay
of the scalar curvature. In particular, no decay of the Ricci curvature is
assumed. In our second result we establish that a local volume ratio condition,
when combined with negativity of the scalar curvature at infinity, is
sufficient for existence of a solution. Our volume ratio condition appears
tight. This paper is based on the DPhil thesis of the first author.
