On rectifiable measures in Carnot groups: representation

TL;DR

This paper advances the understanding of rectifiability in Carnot groups by establishing equivalences between different definitions, deriving a geometric area formula, and proving intrinsic rectifiability of certain sets and spheres.

Contribution

It proves the equivalence of infinitesimal and global rectifiability definitions in Carnot groups and establishes a new geometric area formula for intrinsically differentiable graphs.

Findings

Infinitesimal and global rectifiability are equivalent in Carnot groups.

A new geometric area formula extends previous results.

Almost all preimages of certain Lipschitz functions are intrinsically $C^1$-rectifiable.

Abstract

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $\mathscr{P}$-rectifiable measure. First, we show that in arbitrary Carnot groups the natural \textit{infinitesimal} definition of rectifiabile measure, i.e., the definition given in terms of the existence of \textit{flat} tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with \textit{flat} Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $C^1$-rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $C^1$-rectifiable.