Rickman rugs and intrinsic bilipschitz graphs

Abstract

This paper studies the geometry of bilipschitz maps $f:\mathbb{W}\to \mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W}\subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images $f(\mathbb{W})$ of such maps are called Rickman rugs in the Heisenberg group. The main theorem states that a Rickman rug in the Heisenberg group admits a corona decomposition by intrinsic bilipschitz graphs. As a corollary, Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs. Here, an intrinsic bilipschitz graph is an intrinsic Lipschitz graph, which is simultaneously a Rickman rug. General intrinsic Lipschitz graphs need not be Rickman rugs, even locally, by an example of Bigolin and Vittone.

arXiv comment65 pages, 3 figures

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