TY - JOUR
T1 - Conformal grushin spaces
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2016 American Mathematical Society.
PY - 2016
Y1 - 2016
N2 - 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 )-βdsE 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.
AB - 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 )-βdsE 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.
KW - Alexandrov space
KW - Bi-Lipschitz embedding
KW - Conformal mapping
KW - Grushin plane
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U2 - 10.1090/ecgd/292
DO - 10.1090/ecgd/292
M3 - Article
AN - SCOPUS:85008193331
VL - 20
SP - 97
EP - 115
JO - Conformal Geometry and Dynamics
JF - Conformal Geometry and Dynamics
IS - 6
ER -