Conformal Grushin spaces

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 $ds=d_E(\cdot ,Y{)}^{-\beta }d{s}_E$ for a closed nonempty subset $Y\subset {\mathbb{R}}^n$ and $\beta \in [0,1)$. 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.

arXiv comment20 pages