Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type (Analysis in Theory and Applications, 2024)
We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniquenessresult concerns those such manifolds which are asymptotically locally hyperbolic. Inthis context, our result requires only a partial $C^2$ decay of the metric, namely the fulldecay of the metric in $C^1$ and the decay of the scalar curvature. In particular, no decayof the Ricci curvature is assumed. In our second result we establish that a local volumeratio condition, when combined with negativity of the scalar curvature at infinity, issufficient for existence of a solution. Our volume ratio condition appears tight. Thispaper is based on the DPhil thesis of the first author.
Keywords: Yamabe problem; non-compact manifolds; negative curvature; asymptotically locally hyperbolic; asymptotically warped product; relative volume comparison; non-smooth conformal compactification.