Lipschitz Carnot-Carath\'eodory structures and their limits

TL;DR

This paper establishes conditions under which Lipschitz Carnot-Carathéodory distances converge when the underlying structures of vector fields and norms vary, with applications to subFinsler and Lie group geometries.

Contribution

It proves convergence of Carnot-Carathéodory distances under mild assumptions and provides examples including a subFinsler Mitchell's Theorem and Lie group applications.

Findings

Distances converge locally uniformly under mild assumptions.

Convergence may fail to be uniform if the limit distance is not boundedly compact.

Applications include subFinsler geometry and Lie group structures.

Abstract

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.