Currents in Heisenberg groups

2025/11/24 by Franchi, Bruno, Pansu, Pierre
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2511.18895

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.

Citations

• Flat compactness of normal currents, and charges in Carnot groups
• Metric currents, differentiable structures, and Carnot groups
• Lipschitz extensions into Jet space Carnot groups
• Normal and Integral Currents

Related

• Sub-Riemannian Currents and Slicing of Currents in the Heisenberg group ℍⁿ
• Insight in the Rumin Cohomology and Orientability Properties of the\n Heisenberg Group
• Metric currents and polylipschitz forms
• Metric Currents and Alberti representations
• Metric currents, differentiable structures, and Carnot groups