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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Regularity properties of spheres in homogeneous groups
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by Enrico Le Donne and Sebastiano Nicolussi Golo PDF
Trans. Amer. Math. Soc. 370 (2018), 2057-2084 Request permission

Abstract:

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more detail the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps.
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Additional Information
  • Enrico Le Donne
  • Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, 40014 Jyväskylä, Finland
  • MR Author ID: 867590
  • Email: enrico.ledonne@gmail.com
  • Sebastiano Nicolussi Golo
  • Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, 40014 Jyväskylä, Finland
  • MR Author ID: 1047689
  • Email: sebastiano2.72@gmail.com
  • Received by editor(s): September 21, 2015
  • Received by editor(s) in revised form: March 4, 2016, and July 22, 2106
  • Published electronically: September 7, 2017
  • Additional Notes: The first author was supported by the Academy of Finland project No. 288501
    The second author was supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement No. 607643.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 2057-2084
  • MSC (2010): Primary 28A75, 22E25, 53C60, 53C17, 26A16
  • DOI: https://doi.org/10.1090/tran/7038
  • MathSciNet review: 3739202