Conformal transformation of uniform domains under weights that depend on distance to the boundary

The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary...

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Bibliographic Details
Main Authors Gibara, Ryan, Shanmugalingam, Nageswari
Format Journal Article
LanguageEnglish
Published 04.04.2022
Subjects
Mathematics - Metric Geometry
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Summary:The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.
DOI:10.48550/arxiv.2204.01920