Regularity properties of spheres in homogeneous groups

TL;DR

This paper investigates the regularity of spheres in homogeneous groups, especially Carnot groups like the Heisenberg group, establishing conditions under which these spheres are Lipschitz boundaries and analyzing their geometric properties.

Contribution

It provides new criteria for the Lipschitz regularity of metric spheres in homogeneous groups and extends regularity results to sub-Finsler manifolds without abnormal geodesics.

Findings

Geodesic distances are Lipschitz continuous in sub-Finsler manifolds without abnormal geodesics.

Algebraic criteria are identified for the regularity of homogeneous distances.

Examples of groups with non-smooth, cusped spheres are presented.

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 homogeneous distances on the Heisenberg group.In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres presents cusps.