Metric rectifiability of $\mathbb{H}$-regular surfaces with Hölder continuous horizontal normal

Abstract

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups ${\mathbb{H}}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of $\mathbb{H}$-regular surfaces. We prove that $\mathbb{H}$-regular surfaces in ${\mathbb{H}}^n$ with $\alpha$-Hölder continuous horizontal normal, $\alpha >0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli-Le Donne, where the same conclusion was obtained for $C^{\infty }$-surfaces. In ${\mathbb{H}}^1$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $\epsilon$ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between "big pieces" of metric spaces.

46 pages. v5: incorporated referee suggestions

References in corpus (2)

• Riesz transform and vertical oscillation in the Heisenberg group
• Semi-uniform domains and the $A_{\infty }$ property for harmonic measure

Cited by in corpus (2)

• Lipschitz graphs and currents in Heisenberg groups
• Rickman rugs and intrinsic bilipschitz graphs