Weak contact equations for mappings into Heisenberg groups Zoltan BaloghPiotr HajlaszKevin Wildrick 49Q1553C1746E3553C23Heisenberg groupunrectifiabilitygeometric measure theoryGromov conjectureSobolev mappingsSard theoremsymplectic form Let $k>n$ be positive integers. We consider mappings from a subset of $\mathbb{R}^k$ to the Heisenberg group $\mathbb{H}^n$ with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group $\mathbb{H}^n$ is purely $k$-unrectifiable. We also prove that, for an open set $\Omega \subset \mathbb{R}^k$, the rank of the weak derivative of a weakly contact mapping in the Sobolev space $W^{1,1}_{\mbox{\scriptsize loc}}(\Omega;\mathbb{R}^{2n+1})$ is bounded by $n$ almost everywhere, answering a question of Magnani. Finally, we prove that if $f\colon\Omega\to\mathbb{H}^n$ is $\alpha$-H\"older continuous, $\alpha>\frac{1}{2}$, and locally Lipschitz when considered as a mapping into $\mathbb{R}^{2n+1}$, then $f$ cannot be injective. This result is related to a conjecture of Gromov. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5411 10.1512/iumj.2014.63.5411 en Indiana Univ. Math. J. 63 (2014) 1839 - 1873 state-of-the-art mathematics http://iumj.org/access/