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

Abstract

Two definitions for the rectifiability 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.

Carnot rectifiability and Alberti representations

“…The study of Geometric Measure Theory and rectifiability in Carnot groups was pioneered by the works of Ambrosio-Kirchheim [7], and Franchi-Serapioni-Serra Cassano [40]. The research in this topic has been very active in the last two decades, see, e.g., [60,39,8,59,58,63,45,35,74,34,15,13,14], and the Introduction of [9] for further references. For the sake of exposition, we recall that the definition of rectifiability that we consider in this paper, see Definition 6.1, has been considered and studied by Pauls and Cole-Pauls [69,28].…”

Foliated corona decompositions

“…The function ψ satisfies the intrinsic Lipschitz condition (Definition 2.2); the non-linear nature of this condition is the source of subtleties that ensue (and the reason why basic questions on the rectifiability properties of intrinsic Lipschitz graphs remain open; see e.g. [29]).…”

Intrinsically Lipschitz Graphs on Semidirect Products of Groups

“…in the context of metric spaces. Following this idea, the author introduce other two natural definitions: intrinsically Hölder sections [DD22a] and intrinsically quasi-isometric sections [DD22b] in metric spaces. Yet, following a seminal paper of Cheeger [Che99] (see also [Kei04,KM16]), it is possible to get suitable sets of these sections in order to have convexity and being vector spaces over R and C.…”