Sharp regularity of sub-Riemannian length-minimizing curves
Differential Geometry
2026-01-28 v2 Metric Geometry
Optimization and Control
Abstract
A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a but not length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure. In this paper, we study a class of examples of sub-Riemannian structures that generalizes that presented in [6], and we prove that length-minimizing curves must be at least of class within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp.
Cite
@article{arxiv.2502.00403,
title = {Sharp regularity of sub-Riemannian length-minimizing curves},
author = {Alessandro Socionovo},
journal= {arXiv preprint arXiv:2502.00403},
year = {2026}
}