Regularity properties of spheres in homogeneous groups
Le Donne, E., & Nicolussi Golo, S. (2018). Regularity properties of spheres in homogeneous groups. Transactions of the American Mathematical Society, 370, 2057-2084. https://doi.org/10.1090/tran/7038
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Transactions of the American Mathematical SocietyDate
2018Copyright
© 2017 American Mathematical Society.
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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Academy of Finland; European CommissionFunding program(s)
Academy Research Fellow, AoF; FP7 (EU's 7th Framework Programme)
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about funding
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.License
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