Characteristic points, rectifiability and perimeter measure on stratified groups

Abstract

We establish an explicit formula between the perimeter measure of an open set $E$ with $C^1$ boundary and the spherical Hausdorff measure ${\mathcal{S}}^{Q-1}$ restricted to $\mathrm{der}E$, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to ${\mathcal{S}}^{Q-1}(\mathrm{der}E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The crucial ingredient of this result is the negligibility of “characteristic points” of the boundary. We introduce the notion of “horizontal point”, which extends the notion of characteristic point to arbitrary submanifolds and we prove that the set of horizontal points of a $k$-codimensional submanifold is ${\mathcal{S}}^{Q-k}$-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely, $(\mathbb{G},{\mathbb{R}}^k)$-rectifiability and we prove that Euclidean $k$-codimensional rectifiable sets are $(\mathbb{G},{\mathbb{R}}^k)$-rectifiable.