Currents in Heisenberg groups

TL;DR

This paper compares three different theories of currents adapted to the geometry of Heisenberg groups, showing their equivalence in low dimensions and divergence in higher dimensions, introducing a new class of currents.

Contribution

It demonstrates the equivalence of three current theories in low dimensions and introduces oblique currents, a new class in higher dimensions within Heisenberg groups.

Findings

The three theories coincide in dimensions less than half the ambient dimension.

Ambrosio-Kirchheim currents vanish beyond middle dimension.

Rumin currents correspond to oblique currents in higher dimensions.

Abstract

There are three approaches to currents tuned to the anisotropic geometry of Heisenberg groups: Ambrosio and Kirchheim's approach valid for general metric spaces; distributions dual to horizontal differential forms; distributions dual to Rumin's complex. It is shown that, in dimensions less than half the ambient dimension, these three theories coincide. On the other hand, they diverge beyond middle dimension: Ambrosio-Kirchheim currents vanish, Rumin currents correspond to a new class of Federer-Fleming currents called oblique currents.