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Hölder-topology of the Heisenberg group

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Abstract

The Heisenberg group is an example of a sub-Riemannian manifold homeomorphic, but not bi-Lipschitz equivalent to the Euclidean space. Its metric is derived from curves which are only allowed to move in so-called horizontal directions. We report on some recent progress in the analysis of the Hölder topology of the Heisenberg group, some related and some unrelated to density questions for Sobolev maps into the Heisenberg group. In particular we describe the main ideas behind a result by Hajłasz, Mirra, and the author regarding Gromov’s conjecture, which is based on the linking number. We do not prove or disprove the Gromov Conjecture.

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Acknowledgements

This text was mainly written while the author was preparing a lecture for the 19th Rencontres d’Analyse at UCLouvain in October 2016. He would like to express his gratitude to UCLouvain and the organizers Pierre Bousquet, Jean Van Schaftingen, Augusto Ponce for the kind invitation and their hospitality.

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  1. Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15260, USA

    Armin Schikorra

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  1. Armin Schikorra
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Schikorra, A. Hölder-topology of the Heisenberg group. Aequat. Math. 94, 323–343 (2020). https://doi.org/10.1007/s00010-020-00701-w

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