The generalized Cartan-Hadamard isoperimetric conjecture

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.

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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: The generalized Cartan-Hadamard isoperimetric conjecture · Cartan-Hadamard conjecture

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VibeMathed record: The generalized Cartan-Hadamard isoperimetric conjecture

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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.