Conformal Grushin spaces

Romney, Matthew

doi:10.1090/ecgd/292

Cited by 10 publications

(9 citation statements)

References 27 publications

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“…Our second remark is that Theorem 3 seems to address the same phenomena as the embedding result by J. Seo [12,Theorem 1.1]. It also seems very plausible that by applying [11,Lemma 2.5] and some standard arguments one can show that Theorem 3-emb is equivalent to Romney's version of Seo's result [11,Theorem 2.3]. We have not verify carefully the validity of this approach and give a direct proof of Theorem 3 instead.…”

Section: It Also Includes Arbitrary Big Balls In Following Classes Of...mentioning

confidence: 59%

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Zolotov

2019

Preprint

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SRA-free spaces is a wide class of metric spaces including finite dimensional Alexandrov spaces of non-negative curvature, complete Berwald spaces of nonnegative flag curvature, Cayley Graphs of virtually abelian groups and doubling metric spaces of non-positive Busemann curvature with extendable geodesics. This class also includes arbitrary big balls in complete, locally compact CAT(k)-spaces (k ∈ R) with locally extendable geodesics, finite-dimensional Alexandrov spaces of curvature ≥ k with k ∈ R and complete Finsler manifolds satisfying the doubling condition.We show that SRA-free spaces allow bi-Lipschitz embeddings in Euclidean spaces. As a corollary we obtain a quantitative bi-Lipschitz embedding theorem for balls in finite dimensional Alexandrov spaces of curvature bounded from below conjectured by S. Eriksson-Bique.The main tool of the proof is an extension theorem for bi-Lipschitz maps into Euclidean spaces. This extension theorem is close in nature with the embedding theorem of J. Seo and may be of independent interest. 2010 Mathematics Subject Classification. 51F99. Key words and phrases. bi-Lipschitz embedding, Alexandrov space. 1 Acknowledgements: I thank my advisor Sergey V. Ivanov for all his ideas, advice and continuous support. The mission of investigating connections between SRA(α)-condition and bi-Lipschitz embeddability into Euclidean spaces was given to me by Alexander Lytchak. I'm very grateful for that. I express my gratitude to Nina Lebedeva for multiple useful discussions. Research is supported by "Native towns", a social investment program of PJSC "Gazprom Neft".

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Section: It Also Includes Arbitrary Big Balls In Following Classes Of...mentioning

confidence: 59%

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Zolotov

2019

Preprint

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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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“…Next, the flat manifold M is decomposed into pieces, and embeddings for each piece are patched together using Lipschitz extension theorems. These decompositions are similar in spirit but were developed independently of arguments by Seo and Romney in [50,55].…”

Section: Outline Of Methodsmentioning

confidence: 95%

Quantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifolds

Eriksson-Bique¹

2018

Geom. Topol.

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We construct bi-Lipschitz embeddings into Euclidean space for bounded diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form R n /Γ, where Γ is a discrete group acting properly discontinuously and by isometries on R n . This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces.In the process we develop tools to prove collapsing theory results using algebraic techniques.30L05, 51F99; 53B20, 53C21, 20H15 arXiv:1507.08211v5 [math.MG]

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Conformal Grushin spaces

Cited by 10 publications

References 27 publications

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spaces

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

Quantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifolds

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