Calculus of Variations and Geometric Measure Theory

V. Magnani

Towards a theory of area in homogeneous groups

created by magnani on 20 Oct 2018
modified on 26 Jul 2019


Published Paper

Inserted: 20 oct 2018
Last Updated: 26 jul 2019

Journal: Calc. Var. Partial Differential Equations
Volume: 58
Number: 3
Pages: 39
Year: 2019
Links: work


A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space plays a central role for the existence of blow-ups. Main applications are area-type formulae for new classes of $C^1$ smooth submanifolds. We also study various classes of distances, showing how their symmetries lead to simpler area and coarea formulas. Finally, we establish the equality between spherical measure and Hausdorff measure on all horizontal submanifolds.

Keywords: spherical measure, sub-Riemannian distance, homogeneous group, submanifold, Hausdorff measure