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arXiv · 0907.3295

Metric differentiation, monotonicity and maps to L^1

Jeff Cheeger·Bruce Kleiner
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Abstract

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the cut metric decomposition to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group.

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Jeff Cheeger, Bruce Kleiner. 2009-07-19. Metric differentiation, monotonicity and maps to L^1. https://doi.org/10.1007/s00222-010-0264-9

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