A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability

Seo, Jeehyeon

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https://hdl.handle.net/2142/24329 Copy

Description

Title

A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability

Author(s)

Seo, Jeehyeon

Issue Date

2011-05-25T15:02:38Z

Director of Research (if dissertation) or Advisor (if thesis)

Tyson, Jeremy T.

Doctoral Committee Chair(s)

Wu, Jang-Mei

Committee Member(s)

• Tyson, Jeremy T.
• D'Angelo, John P.
• Merenkov, Sergiy A.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Bi-Lipschitz
• uniformly perfect
• Coloring map
• Whitney decomposition
• the Grushin plane
• singular sub-Riemannian manifold

Abstract

We characterize uniformly perfect, complete, doubling metric spaces which embed bi-Lipschitzly into Euclidean space. Our result applies in particular to spaces of Grushin type equipped with Carnot-Carath ́eodory distance. Hence we obtain the first example of a sub-Riemannian manifold admitting such a bi-Lipschitz embedding. Our techniques involve a passage from local to global information, building on work of Christ and McShane. A new feature of our proof is the verification of the co-Lipschitz condition. This verification splits into a large scale case and a local case. These cases are distinguished by a relative distance map which is associated to a Whitey-type decomposition of an open subset Ω of the space. We prove that if the Whitney cubes embed uniformly bi-Lipschitzly into a fixed Euclidean space, and if the complement of Ω also embeds, then so does the full space.

Graduation Semester

2011-05

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http://hdl.handle.net/2142/24329

Copyright and License Information

Copyright 2011 Jeehyeon Seo

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