Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

We study existence and uniqueness results for the Yamabe problem on noncompact 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.