Conformal grushin spaces

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Abstract

We introduce a class of metrics on Rn generalizing the classical Grushin plane. These are length metrics defined by the line element ds = dE(., Y )dsE for a closed nonempty subset Y ⊂ Rn and ß ε [0, 1). We prove, assuming a Hölder condition on the metric, that these spaces are quasisymmetrically equivalent to Rn 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.

Original languageEnglish
Pages (from-to)97-115
Number of pages19
JournalConformal Geometry and Dynamics
Volume20
Issue number6
DOIs
StatePublished - 2016

Keywords

  • Alexandrov space
  • Bi-Lipschitz embedding
  • Conformal mapping
  • Grushin plane

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