Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds

TL;DR

This paper investigates the Hausdorff dimension and volume in sub-Riemannian manifolds with singular points, extending existing results to non-equiregular cases and analyzing the impact of singularities on geometric measures.

Contribution

It generalizes the relation between Hausdorff dimension and growth vector to non-equiregular submanifolds and characterizes dimensions and volumes in analytic cases with singular points.

Findings

Hausdorff dimension related to growth vectors in strongly equiregular submanifolds

Characterization of Hausdorff dimension in analytic sub-Riemannian manifolds with singular points

Conditions for finiteness of Hausdorff volume of small balls near singularities

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 TN 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,\rho) 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.