Calculus of Variations and Geometric Measure Theory

A. Agrachev - D. Barilari - U. Boscain

On the Hausdorff volume in sub-Riemannian geometry

created by barilari on 05 Mar 2012


Published Paper

Inserted: 5 mar 2012
Last Updated: 5 mar 2012

Journal: Calc. Var. and PDE
Volume: 43
Number: 3-4
Pages: 355-388
Year: 2012


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.