Katok's entropy rigidity conjecture in dimension at least three
Let be a closed connected Riemannian manifold with strictly negative sectional curvature and geodesic flow on the unit tangent bundle. The variational principle gives , where is normalized Liouville measure. Katok (1982) proved for surfaces that equality forces constant curvature, and proved the higher-dimensional statement within the conformal class of a locally symmetric metric; later work gave only local rigidity near real or complex hyperbolic metrics. Katok conjectured that in every dimension equality of the Liouville entropy and the topological entropy characterizes locally symmetric metrics. Does hold for a negatively curved closed manifold only when is locally symmetric?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Smooth dynamics; geodesic flows in negative curvature
- Posed by
- Anatole Katok
- Year posed
- 1982
- Years open
- 44y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 52 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for a closed connected smooth Riemannian manifold of dimension with strictly negative sectional curvature, normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric (real, complex, quaternionic or Cayley hyperbolic, any scale). When the metric is not locally symmetric, the measure of maximal entropy and Liouville measure are mutually singular. It does not treat nonpositive curvature, Anosov geodesic flows without negative curvature, or the related volume-entropy (Besson-Courtois-Gallot) setting.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026).
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Katok's conjecture. For and strictly negative sectional curvature it states if and only if , that is, the universal cover is a rank-one symmetric space of noncompact type at some scale. This is the conjecture in all dimensions where it was open (Katok proved ). The proof derives global smoothness of the stable and unstable distributions from entropy equality via normal forms, formal symmetry algebras and quantitative realization, then applies Benoist-Foulon-Labourie; it was not refereed. There is no Lean formalization.