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

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Title: A characterization of Bi-Lipschitz embeddable metric spaces in terms of local Bi-Lipschitz embeddability
Author(s): Seo, Jeehyeon
Director of Research: Tyson, Jeremy T.
Doctoral Committee Chair(s): Wu, Jang-Mei
Doctoral Committee Member(s): Tyson, Jeremy T.; D'Angelo, John P.; Merenkov, Sergiy
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(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.
Issue Date: 2011-05-25
URI: http://hdl.handle.net/2142/24329
Rights Information: Copyright 2011 Jeehyeon Seo
Date Available in IDEALS: 2011-05-25
Date Deposited: 2011-05
 

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