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- W2055357403 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-Holder 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." @default.
- W2055357403 created "2016-06-24" @default.
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- W2055357403 date "2013-08-23" @default.
- W2055357403 modified "2023-09-27" @default.
- W2055357403 title "Weak contact equations for mappings into Heisenberg groups" @default.
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