On the Spherical Hausdorff Measure in Step 2 Corank 2 sub-Riemannian Geometry

TL;DR

This paper studies the smoothness properties of the spherical Hausdorff measure in generic corank 2 sub-Riemannian geometries, revealing that it is typically a smooth volume with higher regularity in specific cases.

Contribution

It demonstrates that the spherical Hausdorff measure is always a C^1-smooth volume in generic corank 2 structures and is generically C^2 smooth outside a stratified subset, extending previous work.

Findings

Spherical Hausdorff measure is C^1-smooth in generic corank 2 structures.

It is generically C^2 smooth outside a codimension 7 subset.

For rank 4, the measure is generically C^2.

Abstract

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a previous work by the auhors.