Pansu pullback and exterior differentiation for Sobolev maps on Carnot groups

TL;DR

This paper develops a new method for analyzing Sobolev maps on Carnot groups using a polynomial center-of-mass, leading to improved rigidity results and applications to quasiregular mappings in these groups.

Contribution

It introduces a polynomial-based center-of-mass for measures in Carnot groups, enhancing the analysis of Sobolev mappings and extending results to more general groups.

Findings

Well-defined Buser-Karcher center-of-mass for measures in Carnot groups

Weakened assumptions needed for rigidity and structural results

Extended quasiregular mapping results to all Carnot groups

Abstract

We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from recent papers hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general Carnot groups.