Metric differentiation, monotonicity and maps to L¹

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).