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

Magnani Valentino - MalĂ˝ Jan - Mongodi Samuele

year: 2012
journal: J. Geom. Anal.
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 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.

The paper is available on the cvgmt preprint server.