Calculus of Variations and Geometric Measure Theory

V. Magnani

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

created on 27 Jun 2001
modified on 24 Apr 2004


Published Paper

Inserted: 27 jun 2001
Last Updated: 24 apr 2004

Journal: Manuscripta Math.
Volume: 110
Number: 1
Pages: 55-76
Year: 2003


We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate for the Hausdorff dimension of the characteristic set relative to $\,C^{1,1}$ hypersurfaces in 2-step groups and we prove that it has finite $Q-2$ Hausdorff measure, where $Q$ is the homogeneous dimension of the group.