Sub-Riemannian Currents and Slicing of Currents in the Heisenberg group $\mathbb{H}^n$

TL;DR

This paper develops a theory of currents and their slices in the Heisenberg group, revealing unique properties and challenges that differ from classical Riemannian cases, especially in the context of the first Heisenberg group.

Contribution

It introduces the notion of slices of currents in the Heisenberg group and explores their properties, highlighting differences from Riemannian settings and implications for compactness theorems.

Findings

Some properties of currents are similar to Riemannian cases.

Slices of middle dimension $n$ have unique behaviors in $ H^n$.

The case of $ H^1$ shows a divergence due to dimension coincidence.

Abstract

This paper aims to define and study currents and slices of currents in the Heisenberg group $\mathbb{H}^n$. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension $n$, which opens new challenges and scenarios for the possibility of developing a compactness theorem. Furthermore, this suggests that the study of currents on the first Heisenberg group $\mathbb{H}^1$ diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, $2n-1$, coincides with the middle dimension $n$, which triggers a change in the associated differential operator in the Rumin complex.