## 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).

Original language | English (US) |
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Pages (from-to) | 335-370 |

Number of pages | 36 |

Journal | Inventiones Mathematicae |

Volume | 182 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

## ASJC Scopus subject areas

- General Mathematics

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