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On the Hausdorff volume in sub-Riemannian geometry

TitleOn the Hausdorff volume in sub-Riemannian geometry
Publication TypeJournal Article
Year of Publication2012
AuthorsAgrachev, AA, Barilari, D, Boscain, U
JournalCalculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388

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


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