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arXiv · 1911.01760

Sphericalization and flattening with their applications in quasimetric measure spaces

Qingshan Zhou·Yaxiang Li·Xining Li
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Abstract

The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving properties such as Ahlfors regular and doubling property. By using these properties, we generalize a recent result in \cite{WZ}. We also show that the Loewner condition can be preserved under quasim\"obius mapping between two $Q$-Ahlfors regular spaces. Finally, we prove that the $Q$-regularity of $Q$-dimensional Hausdorff measure of Bourdon metric are coincided with Hausdorff measure of Hamenst\"adt metric defined on the boundary at infinity of a Gromov hyperbolic space.

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Qingshan Zhou, Yaxiang Li, Xining Li. 2019-11-05. Sphericalization and flattening with their applications in quasimetric measure spaces. https://arxiv.org/abs/1911.01760

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