Differential Geometry – Currents in Sub-Riemannian Geometry

Question

Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The main underlying idea of Federer and Flaming's notion of Euclidean current generalizes to metric space thanks to the fact that one finds Bilipschitz maps that allow this sort of generalization.

My question is: what is the main difficulty of building currents in the sub-Riemannian settings? Is there some work on the development of an analogous concept of currents in sub-Riemannian manifolds like Carnot groups or in general stratified Lie groups?

I may guess that, on the negative side, we find important examples of metric spaces where the theory of rectifiable currents does not apply simply because the class of rectifiable currents is very poor. The simplest of these examples is the Heisenberg group, due to the following theorem:

If $(X, d)$ is the Heisenberg group $H_1$ endowed with any left
invariant homogeneous metric, then any $k-$dimensional rectifiable
current is identically $0$ for $k=2, 3 , 4.$

Do there exist more general notions of rectifiable currents in sub-Riemannian geometries?

Answer 1

I want to try to give you an answer, although it will be very sketchy. Take into account that the topic is still unclear to the experts themselves.

The key word is “Rumin currents”: they are defined on all Heisenberg groups (and contact manifolds) starting with a pre-selection of differential forms. So, instead of taking the dual of all k-forms, currents are defined taking the dual of Rumin k-forms.

I think the original paper where “Rumin forms” were introduced is Rumin M., Formes differentielles sur les varietes de contact, J. Diff. Geom. 39, 281-330, 1994.

For example, in the first Heisenberg group, we roughly have:

• 0-currents: (generalisation of) points.
• 1-currents: (generalisation of) horizontal curves.
• 2-currents: (generalisation of) surfaces with finite sub-Riemannian area (Hausdorff measure of dimension 3).
• 3-currents: (generalisation of) volume.

Currents should take into account that:

• in dimension 1 we have less objects than in the Euclidean case (not every curve is horizontal).
• in dimension 2 we have more objects than in the Euclidean case (there are surfaces whose sub-Riemannian Hausdorff dimension is 2, but whose Euclidean Hausdorff dimension is >2).

Rumin currents are able to take these differences into account.

However, I think one should ask: what for?

A theory of rectifiability in Carnot groups is still so under-developed that it is not clear what a generalisation to rectifiable currents should look like.

As far as I know, the only application of Rumin currents is in D. Vittone’s paper (which contains a description of Rumin currents): “Lipschitz graphs and currents in Heisenberg groups”, https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/lipschitz-graphs-and-currents-in-heisenberg-groups/0524E2DEE0F4878D20FF2B0641A48CB4

I hope this can help, at least as a start.