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 $\alpha$-H\"older 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.
Categories: math.CA math.DG math.MG