Calculus of Variations and Geometric Measure Theory

V. Magnani - D. Vittone

An intrinsic measure for submanifolds in stratified groups

created by magnani on 05 May 2006
modified on 15 Dec 2008


Published Paper

Inserted: 5 may 2006
Last Updated: 15 dec 2008

Journal: J. Reine Angew. Math.
Volume: 619
Pages: 203-232
Year: 2008


For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the Carnot-Carathéodory distance, respectively. Our main technical tool is an intrinsic blow-up at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations in the Engel group, we show how our results can be extended to higher step stratified groups, provided the submanifold is sufficiently regular.

Keywords: Hausdorff measure, higher codimensional submanifolds, Carnot-Carathéodory distance, stratified groups