Characteristic points, rectifiability and perimeter measure on stratified groups

We establish an explicit formula between the perimeter measure of a domain with differentiable boundary and the (Q-1)-dimensional spherical Hausdorff measure restricted to its boundary, when the ambient space is a stratified group endowed with a sub-Riemannian structure. The spherical Hausdorff measure is built with respect to an arbitrary homogeneous distance and the integer Q denotes the Hausdorff dimension of the group with respect to its Carnot-Carathéodory distance. Our formula implies that the perimeter measure of a bounded domain with differentiable boundary is less than or equal to the (Q−1)-dimensional spherical Hausdorff measure of its boundary up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The same formula for the perimeter measure also provides an explicit expression for the optimal constants in the reciprocal estimates between perimeter measure and spherical Hausdorff measure. This result relies on two main theorems. The first one is a ``negligibility theorem" for singular points of the boundary, namely, the so-called characteristic points. We generalize this notion to submanifolds of arbitrary codimension k and we prove that the set of characteristic points is negligible with respect to the (Q-k)-dimensional spherical Hausdorff measure. The second one, is a ``blow-up theorem" for the perimeter measure of domains with differentiable boundary. We also provide an intrinsic notion of rectifiability for subsets of higher codimension, namely (G,Rk)-rectifiability. As a byproduct of the negligibility theorem, we show that rectifiable sets of codimension k with respect to the usual notion of rectifiability are (G,Rk)-rectifiable.