A review of the tangent space in sub-Finsler geometry and applications to the failure of the $\mathsf{CD}$ condition

Abstract

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the $\mathsf{CD}$ condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.