Contents Online
Mathematical Research Letters
Volume 24 (2017)
Number 4
Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces
Pages: 1177 – 1203
DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n4.a11
Authors
Abstract
We show that, for all $\alpha \geq 0$, the generalized Grushin plane $\mathbb{G}_{\alpha}$ is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space $\mathbb{R}^{\lfloor \alpha \rfloor +2}$, where $\lfloor \alpha \rfloor$ is the integer part of $\alpha$. The target dimension is sharp. This generalizes a recent result of Wu [J.-M.Wu, “Bilipschitz embedding of Grushin plane in $\mathbb{R}^3$”, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015), no. 2, 633–644].
Keywords
generalized Grushin plane, quasiplane, bi-Lipschitz embedding
2010 Mathematics Subject Classification
Primary 53C17. Secondary 30L05, 30L10.
The second author was supported by the Academy of Finland project 257482.
Received 23 March 2015
Accepted 12 February 2016
Published 9 November 2017