Hausdorff volume in non equiregular sub-Riemannian manifolds. (October 2015)

• Record Type: Journal Article
• Authors: Ghezzi, R.; Jean, F.
• Is Part Of: Nonlinear analysis. Volume 126(2015)
• Page Start: 345
• Page End: 377
• Publication Date: 2015-10
• Subjects: Sub-Riemannian geometry -- Hausdorff measures -- Intrinsic volumes -- Geometric measure theory; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.011
• Languages: English
• ISSNs: 0362-546X

Abstract

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).