Metric Rectifiability of H-regular Surfaces with Hölder Continuous Horizontal Normal

Abstract

Two definitions for the rectifiability of hypersurfaces in Heisenberg groups Hn have been proposed: one based on 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 H-regular surfaces. We prove that H-regular surfaces in Hn with α-Hölder continuous horizontal normal, α>0⁠, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for C∞-surfaces. In H1⁠, we prove a slightly stronger result: every codimension-1 intrinsic Lipschitz graph with an ϵ 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.