Geometry of Grushin spaces

Wu, Jun

doi:10.1215/ijm/1455203157

Cited by 17 publications

(10 citation statements)

References 22 publications

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“…Combined with the Beurling-Ahlfors quasiconformal extension [7], Corollary 1.2 yields the following result. An alternative proof of Corollary 1.2 along with new results on questions of quasisymmetric parametrizability and bi-Lipschitz embeddability of highdimensional Grushin spaces can be found in a recent paper of Wu [23].…”

Section: Introductionmentioning

confidence: 99%

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Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Romney

Vellis

2017

Mathematical Research Letters

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Section: Introductionmentioning

confidence: 99%

“…An alternative proof of Corollary 1.2 along with new results on questions of quasisymmetric parametrizability and bi-Lipschitz embeddability of highdimensional Grushin spaces can be found in a recent paper of Wu [23].…”

Section: Introductionmentioning

confidence: 99%

Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Romney

Vellis

2017

Mathematical Research Letters

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“…In fact, the second author showed that H does not bi-Lipschitz embed in any Hilbert space [Li16]. We also record that bi-Lipschitz embeddability of sub-Riemannian manifolds, and especially of Carnot groups, has been studied by various authors [Sem96,Wu15a,RV17,Wu15b,Rom16].…”

Section: Introductionmentioning

confidence: 70%

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

Chousionis¹,

Li²,

Vellis³

et al. 2018

Preprint

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The Heisenberg group H equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which subsets of H bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant L > 0 such that lines L-bi-Lipschitz embed into R 3 and planes L-bi-Lipschitz embed into R 4 . Moreover, C 1,1 2-manifolds without characteristic points as well as all C 1,1 1-manifolds locally L-bi-Lipschitz embed into R 4 where the constant L is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Korányi spheres bi-Lipschitz embed into R 4 with a uniform constant. Finally, we show that there exists a compact, porous subset of H which does not admit a bi-Lipschitz embedding into any Euclidean space.

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“…Note that the standard Grushin plane does not have locally finite Hausdorff 2-measure. However, in the case when β ∈ (0, 1/2), it was shown in [18] and [23] that the β-Grushin plane is bi-Lipschitz equivalent to the Euclidean plane. In particular, the β-Grushin plane is Ahlfors 2-regular.…”

Section: Gluing a Grushin Half-plane To A Euclidean Half-planementioning

confidence: 99%

Uniformization with Infinitesimally Metric Measures

2021

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We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to $${{\mathbb {R}}}^2$$ R 2 . Given a measure $$\mu $$ μ on such a space, we introduce $$\mu $$ μ -quasiconformal maps$$f:X \rightarrow {{\mathbb {R}}}^2$$ f : X → R 2 , whose definition involves deforming lengths of curves by $$\mu $$ μ . We show that if $$\mu $$ μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $$\mu $$ μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

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Geometry of Grushin spaces

Cited by 17 publications

References 22 publications

Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

Uniformization with Infinitesimally Metric Measures

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