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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Conformal Grushin spaces
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by Matthew Romney PDF
Conform. Geom. Dyn. 20 (2016), 97-115 Request permission


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
  • Email:
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 20 (2016), 97-115
  • MSC (2010): Primary 30L05
  • DOI:
  • MathSciNet review: 3492624