Inventiones mathematicae

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

Metric differentiation, monotonicity and maps to L 1

  • Jeff CheegerEmail author
  • Bruce Kleiner


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).


Heisenberg Group Carnot Group Differentiation Theorem Admissible Family Horizontal Pair 
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© Springer-Verlag 2010

Authors and Affiliations

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

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