Marked length rigidity in negative curvature
Let $M$ be a closed connected smooth manifold of dimension $n \ge 3$, and let $g_1, g_2$ be smooth Riemannian metrics on $M$ of negative sectional curvature. For each metric, every nontrivial free homotopy class $[\alpha]$ of closed curves in $M$ contains a unique closed geodesic; write $L_{g_j}([\alpha])$ for its length. The function $L_{g_j}$ is the marked length spectrum of $g_j$: the marking records which class each length belongs to. Croke and Otal proved that the marked length spectrum determines the metric up to isometry on closed negatively curved surfaces. In dimension at least three, Hamenstädt, using work of Besson, Courtois and Gallot, proved this when one of the two metrics is locally symmetric, and Guillarmou and Lefeuvre when the two metrics are close enough in a suitable $C^k$ topology.
Problem. If $L_{g_1}([\alpha]) = L_{g_2}([\alpha])$ for every nontrivial free homotopy class $[\alpha]$, is $g_1 = F^* g_2$ for some diffeomorphism $F \colon M \to M$ homotopic to the identity?
Source: Burns and Katok (1985), Problem 3.1; see Alvarez, Lowe and Smith (2025), Conjecture 1.9. Stated by AIM — Open Applied Problems for dimension at least three, with the diffeomorphism homotopic to the identity.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, closed geodesics, marked length spectrum, negative curvature
References
@article{burns1985,
author = {Burns, K. and Katok, A.},
title = {Manifolds with non-positive curvature},
journal = {Ergodic Theory Dynam. Systems},
volume = {5},
number = {2},
year = {1985},
pages = {307--317},
doi = {10.1017/S0143385700002935},
note = {In collaboration with W. Ballmann, M. Brin, P. Eberlein and R. Osserman},
}
@article{alvarez2025,
author = {Alvarez, S{\'e}bastien and Lowe, Ben and Smith, Graham Andrew},
title = {Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces},
journal = {J. {\'E}c. polytech. Math.},
volume = {12},
year = {2025},
pages = {1197--1227},
doi = {10.5802/jep.309},
eprint = {2412.14389},
archivePrefix = {arXiv},
}
@article{croke1990,
author = {Croke, Christopher B.},
title = {Rigidity for surfaces of non-positive curvature},
journal = {Comment. Math. Helv.},
volume = {65},
number = {1},
year = {1990},
pages = {150--169},
doi = {10.1007/BF02566599},
}
@article{otal1990,
author = {Otal, Jean-Pierre},
title = {Le spectre marqu{\'e} des longueurs des surfaces {\`a} courbure n{\'e}gative},
journal = {Ann. of Math. (2)},
volume = {131},
number = {1},
year = {1990},
pages = {151--162},
doi = {10.2307/1971511},
}
@article{hamenstadt1999,
author = {Hamenst{\"a}dt, Ursula},
title = {Cocycles, symplectic structures and intersection},
journal = {Geom. Funct. Anal.},
volume = {9},
number = {1},
year = {1999},
pages = {90--140},
doi = {10.1007/s000390050082},
}
@article{besson1995,
author = {Besson, G. and Courtois, G. and Gallot, S.},
title = {Entropies et rigidit{\'e}s des espaces localement sym{\'e}triques de courbure strictement n{\'e}gative},
journal = {Geom. Funct. Anal.},
volume = {5},
number = {5},
year = {1995},
pages = {731--799},
doi = {10.1007/BF01897050},
}
@article{guillarmou2019,
author = {Guillarmou, Colin and Lefeuvre, Thibault},
title = {The marked length spectrum of {A}nosov manifolds},
journal = {Ann. of Math. (2)},
volume = {190},
number = {1},
year = {2019},
pages = {321--344},
doi = {10.4007/annals.2019.190.1.6},
eprint = {1806.04218},
archivePrefix = {arXiv},
}
@article{butt2025,
author = {Butt, Karen},
title = {Quantitative marked length spectrum rigidity},
journal = {Geom. Topol.},
volume = {29},
number = {8},
year = {2025},
pages = {3995--4054},
doi = {10.2140/gt.2025.29.3995},
eprint = {2203.12128},
archivePrefix = {arXiv},
}
@misc{aim-marked-length-rigidity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Marked length rigidity in negative curvature}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/057-marked-length-rigidity.md},
note = {Entry 057 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-00535, "Marked length rigidity in negative curvature", clopen.solutions.