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arXiv · 1510.07591

Conformal Grushin spaces

Matthew Romney
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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 $ds = d_E(\cdot,Y)^{-β}ds_E$ for a closed nonempty subset $Y \subset \mathbb{R}^n$ and $β\in [0,1)$. We prove that, assuming a Hölder condition on the metric, 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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Matthew Romney. 2016-02-15. Conformal Grushin spaces. https://arxiv.org/abs/1510.07591

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