Calculus of Variations and Geometric Measure Theory
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R. Ghezzi - F. Jean

Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds

created by ghezzi on 16 Jan 2013

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Inserted: 16 jan 2013
Last Updated: 16 jan 2013

Year: 2013

Abstract:

This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with T N are constant on N. In this case, we generalize the result in 12, which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p,ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.

Tags: GeMeThNES


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