On low-dimensional uniform rectifiability in Heisenberg groups

TL;DR

This paper establishes a new Carleson-type condition for low-dimensional sets in Heisenberg groups to possess large Lipschitz images, advancing the understanding of uniform rectifiability in sub-Riemannian geometry.

Contribution

It introduces a novel Carleson-type criterion for uniform rectifiability in Heisenberg groups, utilizing corona decompositions by normed spaces.

Findings

New Carleson-type condition for $k$-regular sets in $H^n$

Implications between various notions of quantitative rectifiability

Extension of rectifiability results to low-dimensional sets in $H^n$

Abstract

Refining an earlier result due to Hahlomaa, we provide a new Carleson-type condition for $k$-regular sets in the Heisenberg group $\mathbb{H}^n$ to have big pieces of Lipschitz images of subsets of $\mathbb{R}^k$ for $1\leq k\leq n$. Our approach passes via the corona decompositions by normed spaces, recently introduced by Bate, Hyde, and Schul. Along the way, we prove implications between several notions of quantitative rectifiability for low-dimensional sets in $\mathbb{H}^n$.