Smirnov Decomposition of a Horizontal Vector Charge in the Heisenberg Group

TL;DR

This paper proves a Smirnov decomposition for divergence-free horizontal vector currents in Heisenberg space, linking geometric flow lines with measure-theoretic properties in a sub-Riemannian setting.

Contribution

It provides a direct proof of Smirnov decomposition for Federer-Fleming currents in the horizontal distribution of the Heisenberg group, extending classical results.

Findings

Flow lines generate a family of horizontal curves and measures.

A direct proof of Smirnov decomposition in the Heisenberg setting.

Connects geometric flows with measure-theoretic decompositions.

Abstract

A divergence-free horizontal vector current in Heisenberg space may be viewed as an element of the dual space of horizontal test vector fields. By applying a horizontal Liouville theorem in this setting, the flow lines of such a vector field generate a family of horizontal curves and an associated measure on this collection. In this paper, we provide a direct proof of the Smirnov decomposition for a Federer-Fleming current within the horizontal distribution.