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 \( \cS^{Q-1} \) restricted to \( \der E \), 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 \( \cS^{Q-1}(\der E) \) 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 ``characte\-ristic 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 \( \cS^{Q-k} \)-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely, \( (\G,\R^k) \)-rectifiability and we prove that Euclidean $k$-codimensional rectifiable sets are \( (\G,\R^k) \)-rectifiable.