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Katok's entropy rigidity conjecture in dimension at least three

Theorem 1.1: for a closed connected smooth Riemannian manifold of dimension $n\ge3$ 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.

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Katok's entropy rigidity conjecture in dimension at least three · Katok entropy rigidity conjecture

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VibeMathed record: Katok's entropy rigidity conjecture in dimension at least three

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Katok's conjecture. For $n\ge3$ and strictly negative sectional curvature it states $h_{m_L}(\phi_1)=h_{top}(\phi_1)$ if and only if $\nabla R=0$ , 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 $n=2$ ). 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.

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Katok's entropy rigidity conjecture in dimension at least three

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Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Katok, Entropy and closed geodesics (1982)

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