Variational problems concerning sub-Finsler metrics in Carnot groups

TL;DR

This paper explores the convergence and properties of geodesic distances in Carnot groups, linking variational problems, intrinsic distances, and sub-Finsler metrics to deepen understanding of sub-Riemannian geometry.

Contribution

It establishes equivalences between uniform convergence of distances and $\Gamma$-convergence of variational problems in Carnot groups, and analyzes the relation between intrinsic distances and sub-Finsler metrics.

Findings

Characterization of uniform convergence via $\Gamma$-convergence.

Relation between intrinsic distances and sub-Finsler convex metrics.

Insights into the structure of geodesic distances in Carnot groups.

Abstract

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carath\'{e}odory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of $\Gamma$-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.