Hausdorff volume in non equiregular sub-Riemannian manifolds

TL;DR

This paper investigates the Hausdorff volume in non-equiregular sub-Riemannian manifolds, comparing it with smooth volumes, analyzing its decomposition, and characterizing its singular part through examples.

Contribution

It provides a detailed decomposition of the Hausdorff volume, explores conditions for Radon measures, and characterizes the singular part in non-equiregular sub-Riemannian structures.

Findings

Hausdorff volume decomposes into regular and singular parts

Regular part is not always comparable to smooth volume

Conditions for the regular part to be a Radon measure are identified

Abstract

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).