Lipschitz graphs and currents in Heisenberg groups

Abstract

The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

Intrinsically Hölder sections in metric spaces

“…In this section we prove Theorem 1.3. The proof of this statement is similar to the one of [DDLD22, Theorem 1.4] regarding Lipschitz sections theory (see also [Vit20,Theorem 1.5]). We need to mention that there have been several earlier partial results on extensions of Lipschitz graphs in the context of Carnot groups, as for example in Regarding extension theorems in metric spaces, the reader can see [AP20, LN05, Oht09] and their references.…”

Nowhere differentiable intrinsic Lipschitz graphs

“…If is a normal subgroup, the Rademacher Theorem has been proved for general by Antonelli and Merlo in [2]. Recently, the third‐named author [12] proved that Heisenberg groups (with any splitting) satisfy an intrinsic Rademacher Theorem. The question has been open for a long time if is the Engel group (which has step 3) and (see [1]).…”

Parabolic rectifiability, tangent planes and tangent measures

“…They are defined geometrically in terms of cones. In Heisenberg groups a Rademacher theorem was proved in [FSS11] for codimension 1 sets and recently by Vittone for vertical sets of general dimensions in [Vit22]. But Julia, Nicolussi Golo and Vittone showed in [JNV21] that this is false in some Carnot groups.…”