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 Calc. Var. Partial Differential Equations, 2012, Volume 43, Pages 355–388 (Mi cvpde1)

On the Hausdorff volume in sub-Riemannian geometry

A. Agrachevab, D. Barilarib, U. Boscainc

a MIAN, Moscow, Russia
b SISSA, Trieste, Italy
c CNRS, CMAP Ecole Polytechnique, Paris, France

Abstract: 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 $\mathcal{C}^3$ (and $\mathcal{C}^4$ on every smooth curve) but in general not $\mathcal{C}^5$. 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.

 Funding Agency Grant Number European Research Council 239748 Agence Nationale de la Recherche NT09-504490 DIGITEO CONGEO This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, by the ANR Project GCM, program “Blanche”, project number NT09-504490 and by the DIGITEO project CONGEO.

DOI: https://doi.org/10.1007/s00526-011-0414-y

Bibliographic databases:

Document Type: Article
MSC: 53C17, 58C35