Conformal Grushin spaces

TL;DR

This paper introduces a new class of metrics generalizing the Grushin plane, proves their quasisymmetric equivalence to Euclidean space under certain conditions, and demonstrates bi-Lipschitz embeddability with improved methods.

Contribution

It extends the theory of Grushin-type metrics by establishing quasisymmetric equivalence and bi-Lipschitz embeddability without uniform perfectness assumptions.

Findings

Metrics are quasisymmetrically equivalent to Euclidean space under a H"older condition.

Spaces can be embedded bi-Lipschitzly into Euclidean space.

Alternative proof in 2D using curvature growth bounds.

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 that, assuming a H\"older 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.