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Conformal Geometry and Dynamics

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Conformal Grushin spaces


Author: Matthew Romney
Journal: Conform. Geom. Dyn. 20 (2016), 97-115
MSC (2010): Primary 30L05
Published electronically: May 2, 2016
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Abstract: We introduce a class of metrics on $ \mathbb{R}^n$ generalizing the classical Grushin plane. These are length metrics defined by the line element $ ds = d_E(\cdot ,Y)^{-\beta }ds_E$ for a closed nonempty subset $ Y \subset \mathbb{R}^n$ and $ \beta \in [0,1)$. We prove, assuming a Hölder condition on the metric, that these spaces are quasisymmetrically equivalent to $ \mathbb{R}^n$ and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature.


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Additional Information

Matthew Romney
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
Email: romney2@illinois.edu

DOI: https://doi.org/10.1090/ecgd/292
Keywords: Bi-Lipschitz embedding, Grushin plane, Alexandrov space, conformal mapping
Received by editor(s): October 28, 2015
Received by editor(s) in revised form: January 26, 2016, and February 15, 2016
Published electronically: May 2, 2016
Article copyright: © Copyright 2016 American Mathematical Society