A low rank property and nonexistence of higher dimensional horizontal Sobolev sets

TL;DR

This paper proves a low rank property for Sobolev mappings into the Heisenberg group, showing that maximal rank mappings have images with positive Hausdorff measure, and extends the results to higher dimensions using a novel exterior differentiation technique.

Contribution

It introduces a new low rank property for Sobolev maps into Heisenberg groups and solves a key open question about the measure of their images, extending to higher dimensions.

Findings

Maximal rank Sobolev maps have images with positive Hausdorff measure.

The low rank property constrains the structure of Sobolev mappings into Heisenberg groups.

The method extends to higher-dimensional Heisenberg groups.

Abstract

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of the Heisenberg group. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach differs from the previous ones. Its technical aspect consists in performing an "exterior differentiation by blow-up", where the standard distributional exterior differentiation is not possible. This method extends to higher dimensional Sobolev mappings taking values in higher dimensional Heisenberg groups.