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

TL;DR

This paper introduces a conformal transformation technique for uniform domains in metric spaces, depending on boundary distance, which preserves uniformity while transforming unbounded spaces into bounded ones.

Contribution

It proposes a boundary-dependent conformal deformation for metric spaces that preserves uniformity and transforms unbounded domains into bounded ones.

Findings

The deformation is locally bi-Lipschitz near the boundary.

Uniformity is preserved under the conformal transformation.

Transformed spaces become bounded while maintaining key geometric properties.

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.