## Conformal Grushin spaces

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- by Matthew Romney
- Conform. Geom. Dyn.
**20**(2016), 97-115 - DOI: https://doi.org/10.1090/ecgd/292
- Published electronically: May 2, 2016
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## 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, assuming a Hölder condition on the metric, that 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.## References

- Colleen Ackermann,
*An approach to studying quasiconformal mappings on generalized Grushin planes*, Ann. Acad. Sci. Fenn. Math.**40**(2015), no. 1, 305–320. MR**3310085**, DOI 10.5186/aasfm.2015.4021 - Stephanie Alexander, Vitali Kapovich, and Anton Petrunin.
*Alexandrov geometry*. 2014. - Patrice Assouad,
*Plongements lipschitziens dans $\textbf {R}^{n}$*, Bull. Soc. Math. France**111**(1983), no. 4, 429–448 (French, with English summary). MR**763553**, DOI 10.24033/bsmf.1997 - André Bellaïche,
*The tangent space in sub-Riemannian geometry*, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78. MR**1421822**, DOI 10.1007/978-3-0348-9210-0_{1} - Mario Bonk and Urs Lang,
*Bi-Lipschitz parameterization of surfaces*, Math. Ann.**327**(2003), no. 1, 135–169. MR**2006006**, DOI 10.1007/s00208-003-0443-8 - Martin R. Bridson and André Haefliger,
*Metric spaces of non-positive curvature*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486**, DOI 10.1007/978-3-662-12494-9 - Dmitri Burago, Yuri Burago, and Sergei Ivanov,
*A course in metric geometry*, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR**1835418**, DOI 10.1090/gsm/033 - Michael Christ,
*A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral*, Colloq. Math.**60/61**(1990), no. 2, 601–628. MR**1096400**, DOI 10.4064/cm-60-61-2-601-628 - Noel Dejarnette, Piotr Hajłasz, Anton Lukyanenko, and Jeremy T. Tyson,
*On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target*, Conform. Geom. Dyn.**18**(2014), 119–156. MR**3226622**, DOI 10.1090/S1088-4173-2014-00267-7 - Sylvester Eriksson-Bique,
*Quantitative bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds*, preprint arXiv:1507.08211, 2015. - Bruno Franchi and Ermanno Lanconelli,
*Une métrique associée à une classe d’opérateurs elliptiques dégénérés*, Rend. Sem. Mat. Univ. Politec. Torino**Special Issue**(1983), 105–114 (1984) (French). Conference on linear partial and pseudodifferential operators (Torino, 1982). MR**745979** - F. W. Gehring and O. Martio,
*Lipschitz classes and quasiconformal mappings*, Ann. Acad. Sci. Fenn. Ser. A I Math.**10**(1985), 203–219. MR**802481**, DOI 10.5186/aasfm.1985.1022 - Juha Heinonen,
*Lectures on analysis on metric spaces*, Universitext, Springer-Verlag, New York, 2001. MR**1800917**, DOI 10.1007/978-1-4613-0131-8 - Juha Heinonen,
*Nonsmooth calculus*, Bull. Amer. Math. Soc. (N.S.)**44**(2007), no. 2, 163–232. MR**2291675**, DOI 10.1090/S0273-0979-07-01140-8 - Urs Lang and Conrad Plaut,
*Bilipschitz embeddings of metric spaces into space forms*, Geom. Dedicata**87**(2001), no. 1-3, 285–307. MR**1866853**, DOI 10.1023/A:1012093209450 - William Meyerson,
*The Grushin plane and quasiconformal Jacobians*, arXiv preprint arXiv:1112.0078, 2011. - Roberto Monti and Daniele Morbidelli,
*Isoperimetric inequality in the Grushin plane*, J. Geom. Anal.**14**(2004), no. 2, 355–368. MR**2051692**, DOI 10.1007/BF02922077 - Assaf Naor,
*$L_1$ embeddings of the Heisenberg group and fast estimation of graph isoperimetry*, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1549–1575. MR**2827855** - I. G. Nikolaev,
*Bounded curvature closure of the set of compact Riemannian manifolds*, Bull. Amer. Math. Soc. (N.S.)**24**(1991), no. 1, 171–177. MR**1056559**, DOI 10.1090/S0273-0979-1991-15980-X - Matthew Romney and Vyron Vellis,
*Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces*, Math. Res. Lett., to appear. - Stephen Semmes,
*On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights*, Rev. Mat. Iberoamericana**12**(1996), no. 2, 337–410. MR**1402671**, DOI 10.4171/RMI/201 - S. Semmes,
*Bilipschitz embeddings of metric spaces into Euclidean spaces*, Publ. Mat.**43**(1999), no. 2, 571–653. MR**1744622**, DOI 10.5565/PUBLMAT_{4}3299_{0}6 - Jeehyeon Seo,
*A characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability*, Math. Res. Lett.**18**(2011), no. 6, 1179–1202. MR**2915474**, DOI 10.4310/MRL.2011.v18.n6.a9 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Jussi Väisälä,
*Uniform domains*, Tohoku Math. J. (2)**40**(1988), no. 1, 101–118. MR**927080**, DOI 10.2748/tmj/1178228081 - Jang-Mei Wu,
*Bilipschitz embedding of Grushin plane in $\Bbb R^3$*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**14**(2015), no. 2, 633–644. MR**3410474** - Jang-Mei Wu,
*Geometry of Grushin spaces*, Illinois J. Math.**59**(2015), no. 1, 21–41. MR**3459626**

## Bibliographic Information

**Matthew Romney**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
- MR Author ID: 1021342
- Email: romney2@illinois.edu
- Received by editor(s): October 28, 2015
- Received by editor(s) in revised form: January 26, 2016, and February 15, 2016
- Published electronically: May 2, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**20**(2016), 97-115 - MSC (2010): Primary 30L05
- DOI: https://doi.org/10.1090/ecgd/292
- MathSciNet review: 3492624