### 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) |
---|---|

Pages (from-to) | 335-370 |

Number of pages | 36 |

Journal | Inventiones Mathematicae |

Volume | 182 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

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

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 182, no. 2, pp. 335-370. https://doi.org/10.1007/s00222-010-0264-9

}

TY - JOUR

T1 - Metric differentiation, monotonicity and maps to L1

AU - Cheeger, Jeff

AU - Kleiner, Bruce

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

VL - 182

SP - 335

EP - 370

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 2

ER -