Marked length spectrum rigidity in higher dimensions

Prove that two negatively curved closed Riemannian metrics in dimensions at least three are isometric if and only if their marked length spectra agree.

Background

The Burns–Katok marked length spectrum rigidity conjecture asks whether the marked length spectrum determines a negatively curved metric up to isometry. The paper notes that this conjecture was solved for surfaces, including certain extensions to surfaces with Anosov geodesic flow, and in special higher-dimensional cases such as when one metric is locally symmetric or when the metrics are sufficiently close.

The unresolved part explicitly identified in the paper concerns higher dimensions: the general equivalence between equality of marked length spectra and isometry remains unknown for negatively curved closed manifolds of dimension at least three.

References

It is a long-standing conjecture of Burns and Katok that two negatively curved metrics are isometric if and only if their marked length spectra are equal. Katok proved it when the two metrics are in the same conformal class. Then Croke and Otal independently solved the conjecture for surfaces . Their result was extended recently to surfaces with Anosov geodesic flow by Guillarmou, Lefeuvre and Paternain . The conjecture is still open in higher dimension.

— Local unmarked length spectrum rigidity for hyperbolic surfaces  (2609.29188 - Humbert, 24 Sep 2026) in Section 1, subsection “Setting”