Metric differentiation, monotonicity and maps to L1

Research output: Contribution to journalArticle

Abstract

This is one of a series of papers on Lipschitz maps from metric spaces to L1. Here we present the details of results which were announced in Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math. MG/0611954, Sect. 1. 8): a new approach to the infinitesimal structure of Lipschitz maps into L1, and, as a first application, an alternative proof of the main theorem of Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math. MG/0611954), that the Heisenberg group does not admit a bi-Lipschitz embedding in L1. The proof uses the metric differentiation theorem of Pauls (Commun. Anal. Geom. 9(5):951-982, 2001) and the cut metric description in Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math. MG/0611954) to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group. A quantitative version of this classification argument is used in our forthcoming joint paper with Assaf Naor (Cheeger et al. in arXiv:0910. 2026, 2009).

Original languageEnglish (US)
Pages (from-to)335-370
Number of pages36
JournalInventiones Mathematicae
Volume182
Issue number2
DOIs
StatePublished - 2010

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Lipschitz Map
Heisenberg Group
Monotonicity
Metric
Theorem
Metric space
Lipschitz
Monotone
Subset
Series
Alternatives

ASJC Scopus subject areas

  • Mathematics(all)

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Metric differentiation, monotonicity and maps to L1 . / Cheeger, Jeff; Kleiner, Bruce.

In: Inventiones Mathematicae, Vol. 182, No. 2, 2010, p. 335-370.

Research output: Contribution to journalArticle

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