Sobolev mappings and the Rumin complex

TL;DR

This paper demonstrates that Sobolev mappings on contact manifolds preserve the Rumin complex structure, leading to invariance of the Rumin flat complex under bilipschitz transformations, with broader implications for Carnot groups.

Contribution

It establishes that Pansu pullback induces chain mappings on the Rumin complex for Sobolev maps, extending invariance properties in contact and Carnot group settings.

Findings

Pansu pullback respects the Rumin complex on contact manifolds.

The Rumin flat complex is bilipschitz invariant.

Chain mappings are induced on Carnot groups under certain conditions.

Abstract

We consider contact manifolds equipped with Carnot-Caratheodory metrics, and show that the Rumin complex is respected by Sobolev mappings: Pansu pullback induces a chain mapping between the smooth Rumin complex and the distributional Rumin complex. As a consequence, the Rumin flat complex -- the analog of the Whitney flat complex in the setting of contact manifolds -- is bilipschitz invariant. We also show that for Sobolev mappings between general Carnot groups, Pansu pullback induces a chain mapping when restricted to a certain differential ideal of the de Rham complex. Both results are applications of the Pullback Theorem from our previous paper.