Characteristic points, rectifiability and perimeter measure on stratified groups

Abstract

Abstract.We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q−1 restricted to ∂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 S Q−1 (∂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 S Q−k -negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, called (G, R k )-rectifiability, and we prove that Euclidean k-codimensional rectifiable sets are (G, R k )-rectifiable.

Blow-up of regular submanifolds in Heisenberg groups and applications

“…A first consequence of Theorem 1.1 is an explicit formula to compute the (p+1)-dimensional spherical Hausdorff measure of p-dimensional C 1 submanifolds in the Heisenberg group. In fact, thanks to S p+1 -negligibility of characteristic points proved in [22], Theorem 1.1 along with standard theorems on differentation of measures, immediately give the following result. .…”

An intrinsic measure for submanifolds in stratified groups

“…The validity of d-negligibility (1.5) has been proved in [34], when d(Σ) = Q−k. One can check that in two step stratified groups the formula 2p…”

“…Here we address the reader to some relevant papers [1], [2], [5], [4], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [21], [25], [26], [28], [29], [22], [23], [30], [31], [33], [34], [36], [37], [39], [43], [44], [45], [46] and the reference therein.…”

“…Results of [36] rely on the following facts. The Riemannian surface measure admits an intrinsic blow-up at non-horizontal points and the other points form an S Q−k -negligible subset, according to [34]. As a continuation of [36], we wish to find the intrinsic measure associated with the remaining manifolds, namely, horizontal submanifolds.…”

“…Recall that horizontal points of hypersurfaces coincide with the well known characteristic points, that play an important role in the study of hypersurfaces in stratified groups, [5], [9], [14], [15], [18], [20], [22], [23], [34], [38], [40].…”