Weak contact equations for mappings into Heisenberg groups

TL;DR

This paper investigates weak contact equations for mappings into Heisenberg groups, revealing how these equations restrict mapping behaviors, proving unrectifiability, bounding derivatives, and addressing injectivity related to Gromov's conjecture.

Contribution

It introduces a new geometric technique to analyze weak contact equations, providing elementary proofs of unrectifiability, derivative bounds, and non-injectivity for certain mappings into Heisenberg groups.

Findings

Heisenberg group H^n is purely k-unrectifiable.

The rank of the weak derivative is bounded by n almost everywhere.

Mappings with certain regularity cannot be injective.

Abstract

Let k>n be positive integers. We consider mappings from a subset of k-dimensional Euclidean space R^k to the Heisenberg group 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 H^n is purely k-unrectifiable. We also prove that for an open set U in R^k, the rank of the weak derivative of a weakly contact mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by $n$ almost everywhere, answering a question of Magnani. Finally we prove that if a mapping from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when considered as a mapping into R^{2n+1}, then the mapping cannot be injective. This result is related to a conjecture of Gromov.