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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 A.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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
Rights Information:Copyright 2011 Jeehyeon Seo
Date Available in IDEALS:2011-05-25
Date Deposited:2011-05

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