Metric differentiation, monotonicity and maps to L1

• 成果类型: Article
• 署名作者: Cheeger, Jeff; Kleiner, Bruce
• 署名单位: New York University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0264-9
• 发表日期: 2010
• 页码: 335-370
• 关键词: finite perimeter; fine properties; SPACES; ultraproducts; geometry; sets

摘要

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).