Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary
Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary
复制标题
DOI：
10.1515/agms-2022-0141
复制
发表时间：
2022-01
影响因子：
1
通讯作者：
Ryan Gibara;N. Shanmugalingam
中科院分区：
文献类型：
--
作者：
Ryan Gibara;N. Shanmugalingam
摘要球化过程将欧氏空间转化为紧致球。在本说明中，我们提出了一个变形的局部紧的，可整流的路径连通的，非完整的，无界的度量空间，通过使用共形变形，只依赖于距离的度量空间的边界。这种变形是局部双Lipschitz到其边界附近的原始域，但将空间转换成有界域。我们将证明，如果原度量空间是关于其完备化的一致域，则变换后的空间也是一致域。
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.