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Conformal Geometry and Dynamics

ISSN 1088-4173



Conformal Grushin spaces

Author: Matthew Romney
Journal: Conform. Geom. Dyn. 20 (2016), 97-115
MSC (2010): Primary 30L05
Published electronically: May 2, 2016
MathSciNet review: 3492624
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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
MR Author ID: 1021342

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