%0 Journal Article
%J Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388
%D 2012
%T On the Hausdorff volume in sub-Riemannian geometry
%A Andrei Agrachev
%A Davide Barilari
%A Ugo Boscain
%X 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.
%B Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388
%I SISSA
%G en
%U http://hdl.handle.net/1963/6454
%1 6399
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-02-05T13:55:36Z\r\nNo. of bitstreams: 1\r\n1005.0540v3.pdf: 352986 bytes, checksum: 7d3e71cad3c7ff917dc8769ba4cd96c5 (MD5)
%R 10.1007/s00526-011-0414-y