Entropy equality and local symmetry in negative curvature
For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.
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100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
The conjecture was settled by Katok for surfaces and mostly open in higher dimension. The main project of my thesis was to prove a local version of the conjecture near complex hyperbolic metrics. After this I was planning on attacking the conjecture more globally. Open AI claimed a proof of the…