Metric differentiation, monotonicity and maps to L^1

TL;DR

This paper introduces a new approach to understanding the infinitesimal structure of Lipschitz maps into L^1, providing an alternative proof that the Heisenberg group cannot be bi-Lipschitz embedded in L^1, using metric differentiation and cut metric decomposition.

Contribution

It offers a novel method for analyzing Lipschitz maps into L^1 and revisits the non-embedding of the Heisenberg group with an alternative proof.

Findings

Heisenberg group does not admit a bi-Lipschitz embedding into L^1

New approach simplifies the proof using metric differentiation and cut metric decomposition

Classifies monotone subsets of the Heisenberg group

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.