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Regularity properties of spheres in homogeneous groups

Authors: Enrico Le Donne and Sebastiano Nicolussi Golo
Journal: Trans. Amer. Math. Soc. 370 (2018), 2057-2084
MSC (2010): Primary 28A75, 22E25, 53C60, 53C17, 26A16
Published electronically: September 7, 2017
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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

Sebastiano Nicolussi Golo
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, 40014 Jyväskylä, Finland

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.
Article copyright: © Copyright 2017 American Mathematical Society

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