A ug 2 00 9 METRIC DIFFERENTIATION , MONOTONICITY AND MAPS TO L 1

This is one of a series of papers on Lipschitz maps from metric spaces to L. Here we present the details of results which were announced in [CK06, Section 1.8]: a new approach to the infinitesimal structure of Lipschitz maps into L, and, as a first application, an alternative proof of the main theorem of [CK06], that the Heisenberg group does not admit a bi-Lipschitz embedding in L. The proof uses the metric differentiation theorem of Pauls [Pau01] and the cut metric description in [CK06] 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 [CKN].