Year: 2024
Author: Joseph Hogg, Luc Nguyen
Analysis in Theory and Applications, Vol. 40 (2024), Iss. 1 : pp. 57–91
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-2023-0014
Analysis in Theory and Applications, Vol. 40 (2024), Iss. 1 : pp. 57–91
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Yamabe problem non-compact manifolds negative curvature asymptotically locally hyperbolic asymptotically warped product relative volume comparison non-smooth conformal compactification.