JournalsjemsVol. 8, No. 4pp. 585–609

Characteristic points, rectifiability and perimeter measure on stratified groups

  • Valentino Magnani

    Università di Pisa, Italy
Characteristic points, rectifiability and perimeter measure on stratified groups cover
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Abstract

We establish an explicit formula between the perimeter measure of an open set EE with C1C^1 boundary and the spherical Hausdorff measure \cSQ1\cS^{Q-1} restricted to \derE\der E, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and QQ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of EE is less than or equal to \cSQ1(\derE)\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 kk-codimensional submanifold is \cSQk\cS^{Q-k}-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely, (\G,Rk)(\G,\R^k)-rectifiability and we prove that Euclidean kk-codimensional rectifiable sets are (\G,Rk)(\G,\R^k)-rectifiable.

Cite this article

Valentino Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups. J. Eur. Math. Soc. 8 (2006), no. 4, pp. 585–609

DOI 10.4171/JEMS/68