Continuous velocity of every sub-Riemannian length minimizer
Let $M$ be a smooth manifold, $\Delta \subset TM$ a smooth constant-rank bracket-generating distribution and $g$ a smooth inner product on $\Delta$. A horizontal curve is an absolutely continuous curve with $\dot\gamma(t) \in \Delta_{\gamma(t)}$ almost everywhere, and its length is $\int |\dot\gamma|_g \, dt$. Length minimizers that are projections of normal extremals are smooth; the others, the strictly abnormal minimizers, exist by an example of Montgomery. Hakavuori and Le Donne proved that length minimizers have no corners, and Barilari, Chitour, Jean, Prandi and Sigalotti that for rank-two distributions of step at most four they are of class $C^1$. Chitour, Jean, Monti, Rifford, Sacchelli, Sigalotti and Socionovo found a polynomial structure on $\mathbb{R}^3$ with a length minimizer that is $C^2$ but not $C^3$ at an endpoint, and Rossi, Schiavoni Piazza and Socionovo one that is $C^2$ but not $C^3$ at an interior point.
Problem. Must every horizontal curve $\gamma : [0, L] \to M$ that minimizes length between its endpoints and is parametrized by unit speed belong to $C^1([0, L]; M)$, with one-sided derivatives at the endpoints? This includes strictly abnormal minimizers.
Source: The regularity problem for sub-Riemannian length minimizers (see Montgomery (2002), §10.1); in the C¹ form left open by Chitour, Jean, Monti, Rifford, Sacchelli, Sigalotti and Socionovo (2025), §1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- OC
- Keywords
- abnormal minimizers, AIM Open Applied Problems, sub-Riemannian geodesics
References
@book{montgomery2002,
author = {Montgomery, Richard},
title = {A Tour of Subriemannian Geometries, Their Geodesics and Applications},
series = {Mathematical Surveys and Monographs},
volume = {91},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2002},
doi = {10.1090/surv/091},
}
@misc{chitour2025,
author = {Chitour, Yacine and Jean, Fr{\'e}d{\'e}ric and Monti, Roberto and Rifford, Ludovic and Sacchelli, Ludovic and Sigalotti, Mario and Socionovo, Alessandro},
title = {Not all sub-{R}iemannian minimizing geodesics are smooth},
year = {2025},
eprint = {2501.18920},
archivePrefix = {arXiv},
}
@article{montgomery1994,
author = {Montgomery, Richard},
title = {Abnormal minimizers},
journal = {SIAM J. Control Optim.},
volume = {32},
number = {6},
year = {1994},
pages = {1605--1620},
doi = {10.1137/S0363012993244945},
}
@article{hakavuori2016,
author = {Hakavuori, Eero and Le Donne, Enrico},
title = {Non-minimality of corners in subriemannian geometry},
journal = {Invent. Math.},
volume = {206},
number = {3},
year = {2016},
pages = {693--704},
doi = {10.1007/s00222-016-0661-9},
}
@article{barilari2020,
author = {Barilari, D. and Chitour, Y. and Jean, F. and Prandi, D. and Sigalotti, M.},
title = {On the regularity of abnormal minimizers for rank 2 sub-{R}iemannian structures},
journal = {J. Math. Pures Appl. (9)},
volume = {133},
year = {2020},
pages = {118--138},
doi = {10.1016/j.matpur.2019.04.008},
eprint = {1804.00971},
archivePrefix = {arXiv},
}
@misc{rossi2026,
author = {Rossi, Tommaso and Schiavoni Piazza, Alec Jacopo Almo and Socionovo, Alessandro},
title = {Interior singularity and branching of geodesics in real-analytic sub-{R}iemannian manifolds},
year = {2026},
eprint = {2603.23068},
archivePrefix = {arXiv},
}
@misc{aim-c1-subriemannian-minimizers,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuous velocity of every sub-Riemannian length minimizer}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/124-c1-subriemannian-minimizers.md},
note = {Entry 124 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00506, "Continuous velocity of every sub-Riemannian length minimizer", clopen.solutions.