Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions

TL;DR

This paper explores sub-Finsler geometry through time-optimal control for specific nilpotent distributions, characterizing extremals, and analyzing metric spheres' properties.

Contribution

It introduces a control-theoretic approach to sub-Finsler structures on nilpotent distributions, including non-smooth cases, and studies their geometric properties.

Findings

Characterization of extremal curves for the distributions

Calculation of metric spheres and proof of Euclidean rectifiability

Analysis of optimality conditions in non-smooth, non-strictly convex settings

Abstract

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.