TY - JOUR
T1 - Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type
AU - Hogg , Joseph
AU - Nguyen , Luc
JO - Analysis in Theory and Applications
VL - 1
SP - 57
EP - 91
PY - 2024
DA - 2024/04
SN - 40
DO - http://doi.org/10.4208/ata.OA-2023-0014
UR - https://global-sci.org/intro/article_detail/ata/23020.html
KW - Yamabe problem, non-compact manifolds, negative curvature, asymptotically locally hyperbolic, asymptotically warped product, relative volume comparison, non-smooth conformal compactification.
AB - <p style="text-align: justify;">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.</p>