Hausdorff volume in non equiregular sub-Riemannian manifolds

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 condit...

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Bibliographic Details
Published in:Nonlinear analysis 2015-10, Vol.126, p.345-377
Main Authors: Ghezzi, R., Jean, F.
Format: Article
Language:English
Subjects:
Decomposition
Differential Geometry
Geometric measure theory
Hausdorff measures
Intrinsic volumes
Manifolds
Mathematics
Metric Geometry
Nonlinearity
Optimization and Control
Radon
Sub-Riemannian geometry
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Summary: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).
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2015.06.011