Universal volume growth bounds from positive intermediate curvature
In 1986 Gromov asked whether every complete -dimensional Riemannian manifold with and satisfiesfor every and every . The three-dimensional case had been settled, and higher dimensions were known only under extra hypotheses such as nonnegative sectional curvature, noncollapsing or an injectivity-radius bound.
This paper answers the question affirmatively, as the case of a uniform family: for every , if and the -intermediate curvature of Brendle-Hirsch-Johne is at least 1, then . At this gives linear volume growth under positive biRicci curvature in every dimension.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Differential geometry
- Posed by
- Mikhail Gromov
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-08-13
- Model
- GPT-5.6 Sol
- Vendor
- OpenAI
- Collaborators
- Gioacchino Antonelli
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 38 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Gromov's 1986 question has two independent proofs one day apart, and the first one public is not the AI-assisted one. Jian Ge posted "Heat kernel geometry and Gromov's volume growth conjecture" to arXiv on 13 August 2026, proving the same theorem by a different route - heat-kernel Fisher metric and Nash entropy. This paper appeared on 14 August, its author stating he learned of Ge's work only after his own manuscript was complete. The headline axes record this paper because it is the one with an AI in the loop.
What it adds beyond Gromov's case is the uniform family: for every , nonnegative Ricci plus a positive lower bound on the -intermediate curvature forces at most -dimensional volume growth. That interpolates between Ricci-type and scalar hypotheses, gives linear growth under positive biRicci curvature in every dimension at , and yields a noncollapsed Urysohn-width bound. The author states this extension is his own contribution, not the model's.
What the AI did
From the paper's own disclosure of AI tools. The work made substantial use of OpenAI's GPT-5.6 Sol at Ultra reasoning effort. GPT proposed the central inductive procedure, based on the Hodge obstruction and rank improvement, behind the proof of Theorem 1.1 - the scalar-curvature case, which is Gromov's question. Antonelli formulated and guided the problem, suggested strategies and literature, and developed the note from that strategy.
He is explicit about how far the published proof moved from the model's: it is effective, where the suggested strategy proceeded by contradiction, and it is much less reliant on Kapovitch-Wilking's Theorem 5.1 - both changes emerging from discussions with Daniele Semola, Elia Bruè and Kai Xu. He also states that the extension to the full Brendle-Hirsch-Johne intermediate-curvature family, which is the paper's general theorem, is his own contribution and not the model's.
Verification
A five-day-old arXiv preprint, unrefereed and not formally endorsed, so this stays Unreviewed. It is not unexamined, though: the author thanks Elia Bruè, Otis Chodosh, Alessandro Cucinotta, Chao Li, Aaron Naber, Daniele Semola and Kai Xu for comments on preliminary versions, and the acknowledgments record two specific ways their comments changed the argument. That is a stronger signal than most preprints of this age carry, but comments are not endorsement, and this site ran no independent check of its own.
The strongest external evidence is indirect and worth stating: Jian Ge's independent preprint of 13 August reaches the same conclusion by heat-kernel Fisher metric and Nash entropy, methods with nothing in common with the splitting-map and Hodge-obstruction argument here. Two unrelated routes to the same statement, a day apart, is meaningful corroboration of the statement even though neither proof has been checked.
Sources
Submitted by VibeGene on