The generalized Cartan-Hadamard isoperimetric conjecture

Let $(M^n,g)$ be a Cartan-Hadamard manifold: complete, simply connected, with sectional curvature $\le\kappa\le0$ . The Euclidean Cartan-Hadamard conjecture (usually attributed to Aubin, also to Gromov and to Burago-Zalgaller) asserts that every bounded smooth domain $\Omega\subset M$ satisfies the Euclidean isoperimetric inequality $|\partial\Omega|\ge n\omega_n^{1/n}|\Omega|^{(n-1)/n}$ , with equality only for regions isometric to Euclidean balls. Its generalized form replaces the right side by the boundary area of the equal-volume geodesic ball in the simply connected space form of curvature $\kappa$ . It was known in dimension two (Weil, Beckenbach-Rado, Bol), three (Kleiner) and, for $\kappa=0$ , four (Croke). Does the comparison with the model ball of curvature $\kappa$ hold in every dimension?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Riemannian geometry; isoperimetric inequalities
Posed by: Thierry Aubin (Euclidean form, 1976); the curvature-kappa generalization is usually credited to Gromov and to Burago-Zalgaller
Year posed: 1976
Years open: 50y
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 $n\ge2$ , $\kappa\le0$ and $(M^n,g)$ complete simply connected with $\mathrm{Sec}\le\kappa$ , every measurable $E$ of finite volume and finite perimeter has $P(E)\ge\mathcal I_{\sqrt{-\kappa}}(\mathrm{vol}(E))$ , the model-ball profile. Theorem 1.2: for $\mathrm{Sec}\le0$ , a bounded positive-volume set attains the Euclidean bound iff it agrees a.e. with an open region whose induced metric is a round Euclidean ball (the ambient manifold need not be flat). Corollaries give Faber-Krahn and Saint-Venant comparisons with the model ball. Equality rigidity for $\kappa<0$ is not claimed. The companion proves the sharp Euclidean filling bound for integral $n$ -cycles, $n\ge2$ , in any proper CAT(0) space and deduces the Euclidean inequality for domains in dimension $d\ge3$ . Recent preprints cited by the paper (Chen-Ghomi-Wang, dimensions 3 to 9) overlap in low dimensions.

What the AI did

The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, 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, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The CAT(0) filling companion (same date) gives a second, independent route to the Euclidean case in dimension at least three.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the principal manuscript were read against the conjecture; Theorem 1.1 is the generalized comparison for every $n\ge2$ and $\kappa\le0$ , for all finite-volume finite-perimeter sets. The proof was not refereed. The Lean work covers the companion, not the principal manuscript. lean/docs/337.md points to two challenges, ComparatorChallenges/SharpCAT0Filling.json (theorem OAI.CAT0Fillings.sharp_integral_filling, solution module OAI.Geometry.CAT0Fillings.Main) and FillingCoefficient.json (OAI.SharpIntegralFillings.coefficient_optimal, module OAI.Geometry.IntegralFillings.Main); both solution files exist at the pinned commit, and neither challenge is in the formalization catalogue formalization.yaml. The statements were read here: every compactly supported integral $n$ -cycle, $n\ge2$ , in a proper CAT(0) space has a compactly supported integral filling of mass at most $c_n M(T)^{(n+1)/n}$ with the Euclidean coefficient, and that coefficient is optimal. This is the Euclidean case in current form; the passage to domains in a manifold, the curvature- $\kappa$ comparison and the equality rigidity are not formalized. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statements cover only the companion paper's Euclidean CAT(0) filling bound.

The generalized Cartan-Hadamard isoperimetric conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion