Browse › Differential geometry › 337 Sharp Cartan–Hadamard isoperimetry and rigidity
Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity
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Abstract
We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 337: Sharp Cartan–Hadamard isoperimetry and rigidity (release Lean docs, 2026). The scope covers only the companion paper's sharp CAT(0) integral-filling bound (cycle dimensions at least two) and the optimality of its Euclidean coefficient; no Lean statement of this paper's Theorems 1.1–1.2 (model-profile comparison for $\kappa\le0$, equality rigidity) is listed. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Shibing Chen, Mohammad Ghomi, Peng Wang, The Cartan-Hadamard conjecture in dimension five (arXiv:2609.11005, 10 Sep 2026). An earlier preprint by the authors of the cited arXiv:2609.20517, proving the Euclidean comparison in dimension 5 and giving a new dimension-3 proof via boundary-pair integrals, Jacobi-field estimates and Kleiner's profile argument; it bears on the overlap discussed in the paper's "History and related work" and is not in the bibliography.
- Mohammad Ghomi, Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity (arXiv:2605.24638, 23 May 2026). Proves the total Gauss–Kronecker curvature bound, and hence the Euclidean isoperimetric inequality, for Cartan–Hadamard manifolds with nullity index at least $n-3$, extending the Ghomi–Spruck total-curvature route the paper cites; it is not in the bibliography.
Digests and expositions
- Latent Seminar, Cartan–Hadamard isoperimetry and sharp integral fillings (7 October 2026). This exposition outlines both manuscripts' claimed proof mechanisms, including the hypersurface Sobolev estimate, confined profiles, equality rigidity and the CAT(0) variance argument, while explicitly disclaiming independent verification of the complete proofs. [family-level]
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit report, 10 October 2026). The report lists family 337 among families whose headline includes a result from a manuscript outside the scope note's linked formalization, specifically identifying the generalized smooth isoperimetric comparison. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 337
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026,
author = {{OpenAI}},
title = {{Generalized Cartan--Hadamard isoperimetry and Euclidean equality rigidity}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf}{OAI:Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/337.md. The Lean statements have not been checked by people against the paper's claims.
Family 337: Sharp Cartan–Hadamard isoperimetry and rigidity
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT$(0)$ spaces.
Other papers in this family:
- Sharp integral fillings in CAT(0) spaces · September 23, 2026