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arXiv : 1005.0540

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On the Hausdorff volume in sub-Riemannian geometry

Andrei Agrachev 1, Davide Barilari 1, Ugo Boscain 2

(15/04/2011)

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.

1 :	Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS)
	Scuola Internazionale Superiore di Studi Avanzati/International School for Advanced Studies (SISSA/ISAS)
2 :	Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
	Polytechnique - X – CNRS : UMR7641

Domaine

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Mathématiques/Géométrie différentielle Mathématiques/Analyse fonctionnelle

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Contributeur : Davide Barilari <>

Soumis le : Lundi 20 Février 2012, 20:54:16

Dernière modification le : Lundi 20 Février 2012, 20:54:16