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arXiv · 2204.01920

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

Ryan Gibara·Nageswari Shanmugalingam
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Abstract

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.

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Ryan Gibara, Nageswari Shanmugalingam. 2022-04-05. Conformal transformation of uniform domains under weights that depend on distance to the boundary. https://arxiv.org/abs/2204.01920

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