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Inventiones mathematicae

, Volume 182, Issue 2, pp 335–370 | Cite as

Metric differentiation, monotonicity and maps to L 1

  • Jeff CheegerEmail author
  • Bruce Kleiner
Article

Abstract

This is one of a series of papers on Lipschitz maps from metric spaces to L 1. 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 L 1, 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 L 1. 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).

Keywords

Heisenberg Group Carnot Group Differentiation Theorem Admissible Family Horizontal Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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