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

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arxiv 1711.03533 v1 pith:AFCCA4PT submitted 2017-11-09 math.MG math.CA

Guy C. David , Kyle Kinneberg This is my paper · ORCID
classification math.MGmath.CA
keywords bi-lipschitzmapscarnotspacesgroupslipschitzmanyahlfors
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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.

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Cited by 1 Pith paper

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  1. On the Lipschitz dimension of Cheeger-Kleiner

    math.MG 2019-08 conditional novelty 8.0 of 10

    This paper proves that non-abelian Carnot groups have infinite Lipschitz dimension and computes the Lipschitz dimension of snowflakes, trees, buildings, and Sierpinski carpets.

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