Matthew Romney — Page 2

arXiv2016

Conformal Grushin spaces

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.

math.MG↗

arXiv2016

Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

math.MG↗