Conformal Grushin spaces

Romney, Matthew

doi:10.1090/ecgd/292

Conformal Grushin spaces
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by Matthew Romney

Conform. Geom. Dyn. 20 (2016), 97-115

DOI: https://doi.org/10.1090/ecgd/292

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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