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Regularity properties of spheres in homogeneous groups. (English) Zbl 1385.53017
For a step-$$r$$ Carnot group, it is a well-known result of Nagel, Stein, and Wainger that the sub-Riemannian geodesic metric is locally $$\frac{1}{r}$$-Hölder. The authors are interested in criteria for which the distance function is actually Euclidean Lipschitz away from the diagonal. The authors study this problem for sub-Finsler manifolds, the Carnot groups of which form a prominent subclass. In Theorem 1.1, the authors prove that if $$G$$ is a stratified manifold with a sub-Finsler metric $$d$$ and $$p\in G$$ is such that geodesics leading to it from the origin are regular, then the function $$d_0 = d(0,\cdot)$$ is locally Lipschitz with respect to any Riemannian metric on $$G$$. With the same assumptions as in Theorem 1.1, the authors prove that if $$p$$ is on the unit sphere, then the unit sphere is locally a Lipschitz graph around $$p$$. Their proofs rely on using the weak* topology of the control functions to bound the pointwise Lipschitz constant of $$d_0$$ and studying properties of the end-point map on sub-Finsler manifolds. The authors conclude by proving Lipschitz properties about the unit ball and unit sphere of the Heisenberg group with respect to any homogeneous distance.

##### MSC:
 53C17 Sub-Riemannian geometry 28A75 Length, area, volume, other geometric measure theory 22E25 Nilpotent and solvable Lie groups 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 26A16 Lipschitz (Hölder) classes
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