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 C2
decay of the metric, namely the full decay of the metric in C1 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.Comment: To appear in Analysis in Theory and Application