Currents in Heisenberg groups
Bruno Franchi, Pierre Pansu
TL;DR
This work unifies three anisotropic current theories in Heisenberg groups by establishing precise correspondences: for k ≤ n, Rumin currents, FF horizontal currents, and Ambrosio–Kirchheim metric currents align; for k > n, Rumin/oblique currents prevail, with oblique mass matching Rumin mass and diverging from metric currents. The oblique viewpoint arises from Rumin’s complex and yields a robust mass notion under dilation, offering a rich supply of oblique currents beyond the horizontal regime. A random deformation theorem and deformation techniques are developed to connect FF currents with metric currents, enabling a deep structural comparison and integral-current correspondences across frameworks. The results provide a coherent toolkit for geometric measure theory in sub-Riemannian settings, with explicit mass relations, boundary compatibility, and constructive pathways between theories. This has potential implications for studying submanifold-like objects and currents in Heisenberg-type spaces, especially beyond the middle dimension where new oblique phenomena emerge.
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.