Lipschitz and bi-Lipschitz maps from PI spaces to Carnot groups

TL;DR

This paper investigates bi-Lipschitz properties of Lipschitz maps from PI spaces to Carnot groups, proving the existence of bi-Lipschitz tangents and conditions for bi-Lipschitz decompositions, confirming a conjecture of Semmes.

Contribution

It establishes the presence of many bi-Lipschitz tangents in such maps and characterizes when these maps can be decomposed into bi-Lipschitz pieces, especially for Euclidean targets.

Findings

Maps have many bi-Lipschitz tangents, confirming Semmes' conjecture.

Bi-Lipschitz decomposition exists when the target is Euclidean.

For general Carnot groups, decomposition depends on the geometry of the image set.

Abstract

This paper deals with the problem of finding bi-Lipschitz behavior in non-degenerate Lipschitz maps between metric measure spaces. Specifically, we study maps from (subsets of) Ahlfors regular PI spaces into sub-Riemannian Carnot groups. We prove that such maps have many bi-Lipschitz tangents, verifying a conjecture of Semmes. As a stronger conclusion, one would like to know whether such maps decompose into countably many bi-Lipschitz pieces. We show that this is true when the Carnot group is Euclidean. For general Carnot targets, we show that the existence of a bi-Lipschitz decomposition is equivalent to a condition on the geometry of the image set.