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.
|Titolo:||On the Hausdorff volume in sub-Riemannian geometry|
|Autori:||AGRACHEV A.; BARILARI D; BOSCAIN U.|
|Rivista:||CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS|
|Data di pubblicazione:||2012|
|Digital Object Identifier (DOI):||10.1007/s00526-011-0414-y|
|Appare nelle tipologie:||1.1 Journal article|