Calculus of Variations and Geometric Measure Theory

G. Antonelli - E. Le Donne - S. Nicolussi Golo

Lipschitz Carnot-Carathéodory structures and their limits

created by ledonne on 15 Nov 2021
modified by antonelli on 13 Aug 2022


Accepted Paper

Inserted: 15 nov 2021
Last Updated: 13 aug 2022

Journal: Journal of Dynamical and Control Systems
Year: 2021

ArXiv: 2111.06789 PDF


In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carath\'eodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell's Theorem with continuously varying norms, and a general convergence result for Carnot-Carath\'eodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

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