Towards a theory of area in homogeneous groups

TL;DR

This paper develops a general framework for computing measures of submanifolds in homogeneous groups, introducing new formulas and establishing measure equivalences, with applications to smooth submanifolds and distances.

Contribution

It provides a unified approach to measure computation in homogeneous groups, including new area formulas and measure equivalences for various classes of submanifolds.

Findings

Established area-type formulas for $C^1$ smooth submanifolds.

Proved the equality of spherical and Hausdorff measures on horizontal submanifolds.

Analyzed the impact of distance symmetries on measure formulas.

Abstract

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.