A Low Rank Property and Nonexistence of Higher-Dimensional Horizontal Sobolev Sets

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 \(\mathbb {H}^1\), and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of \(\mathbb {H}^1\). This provides a complete solution to a question raised in a paper by Balogh et al. (Ergodic Theory Dynam Syst 26(3):621–651, 2006). 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.