TY - JOUR

T1 - Metric differentiation, monotonicity and maps to L1

AU - Cheeger, Jeff

AU - Kleiner, Bruce

N1 - Funding Information:
Research supported in part by NSF grant DMS-0805939.
Funding Information:
Research supported in part by NSF grant DMS-0704404.

PY - 2010

Y1 - 2010

N2 - 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).

AB - 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).

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U2 - 10.1007/s00222-010-0264-9

DO - 10.1007/s00222-010-0264-9

M3 - Article

AN - SCOPUS:77957967799

VL - 182

SP - 335

EP - 370

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 2

ER -