Let (M,g) be a closed Riemannian 3-manifold, let R_g denote its scalar curvature, and let h(g) denote its volume entropy. Agol–Storm–Thurston conjecture. If R_g ≥ -6, then h(g)≤ 2. This conjecture proposes a direct relation between scalar curvature and volume entropy in dimension three. The paper's abstract states that its construction gives counterexamples, so the conjecture is refuted.

Agol–Storm–Thurston scalar curvature and volume entropy conjecture

The Agol–Storm–Thurston conjecture asserts that every closed Riemannian $3$ -manifold with scalar curvature at least $-6$ has volume entropy at most $2$ .

December 2023 counterexample claim

Demetre Kazaras, Antoine Song, and Kai Xu claim that every closed hyperbolic $3$ -manifold admits a metric with $R_g\geq -6$ and $h(g)>2$ , refuting the conjecture. Their construction uses a drawstring deformation; the resulting metrics can have arbitrarily large entropy or be arbitrarily close to the hyperbolic metric in the $C^0$ topology. The paper states that the analogous questions for metrics $C^\infty$ -close to hyperbolic metrics or for negatively curved metrics remain open.

Current status (as of September 2026): The conjecture is claimed refuted by the Kazaras–Song–Xu construction, but this report records the claim as unverified; the stated restricted cases remain open.

Agol–Storm–Thurston scalar curvature and volume entropy conjecture

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