TY - JOUR
T1 - On the Hausdorff volume in sub-Riemannian geometry
JF - Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388
Y1 - 2012
A1 - Andrei A. Agrachev
A1 - Davide Barilari
A1 - Ugo Boscain
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6454
U1 - 6399
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -