  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.
