Local unmarked length spectrum rigidity for negatively curved metrics

Establish that for every negatively curved closed manifold \((M^n,g)\), there exist an integer \(k\) and \(\epsilon>0\) such that any negatively curved metric \(g'\) satisfying \(\|g-g'\|_{C^k}<\epsilon\) has the same unmarked length spectrum as \(g\) if and only if \(g\) and \(g'\) are isometric.

Background

The paper formulates a local analogue of unmarked length spectrum rigidity. The unmarked length spectrum is the multiset of lengths of all closed geodesics, without recording their free homotopy classes. Although equality of marked length spectra is known to imply rigidity in several settings, equal unmarked length spectra can occur for non-isometric hyperbolic metrics globally, as shown by examples attributed to Vignéras.

The conjecture asserts that this non-uniqueness disappears locally around any negatively curved metric: sufficiently close metrics with identical unmarked length spectra should necessarily be isometric. The paper proves this conjecture for hyperbolic metrics on closed surfaces, while the general negatively curved and higher-dimensional formulation remains unresolved.

References

Nevertheless, in view of the recent breakthrough of Guillarmou and Lefeuvre on the local rigidity of the marked length spectrum, it seems reasonable to state the following conjecture. Let $(Mn,g)$ be a negatively curved closed manifold of dimension $n\geq 2$. There exists $k\in \mathbb N$ and $\epsilon>0$ such that for any negatively curved metric $g'$ with $|g-g'|_{Ck}<\epsilon$, one has $L(g)=L(g')$ if and only if $g$ and $g'$ are isometric.

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