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

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'', when the standard distributional exterior differentiation is not possible. This method extends to higher dimensional Sobolev mappings of suitable Sobolev exponents and taking values in higher dimensional Heisenberg groups.