Calculus of Variations and Geometric Measure Theory

V. Magnani

Characteristic points, rectifiability and perimeter measure on stratified groups

created on 22 Jul 2002
modified by magnani on 09 Jan 2007


Published Paper

Inserted: 22 jul 2002
Last Updated: 9 jan 2007

Journal: J. Eur. Math. Soc.
Volume: 8
Number: 4
Pages: 585-609
Year: 2006


We establish an explicit formula between the perimeter measure of a domain $E$ with $C^1$ boundary and the spherical Hausdorff measure $S^{Q-1}$ restricted to its boundary, when the ambient space is a stratified group endowed with a sub-Riemannian structure. The spherical Hausdorff measure $S^{Q-1}$ 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 $E$ with $C^1$ boundary is less than or equal to the $S^{Q-1}$-measure of the boundary of $E$ 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 characte\-ristic points. We gene\-ralize this notion to submanifolds of arbitrary codimension $k$ and we prove that the set of characteristic points is $S^{Q-k}$-negligible. The second one, is a ``blow-up theorem" for the perimeter measure of domains with $C^1$ boundary. We also provide an intrinsic notion of rectifiability for subsets of higher codimension, namely $(G,R^k)$-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 $(G,R^k)$-rectifiable.

Keywords: Rectifiability, stratified groups, perimeter measure, characteristic points