Result 337, Differential geometry

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)(0) spaces.

Lean formalization Proof

The bigger picture

Why it matters

How little boundary can enclose a given volume in a curved space? These manuscripts claim that nonpositive curvature cannot beat the corresponding constant-curvature ball, and extend a related Euclidean benchmark to fillings of generalized surfaces.

What changes?

In every dimension, the first manuscript reports that a complete, simply connected smooth manifold with sectional curvature at most a nonpositive constant kappa obeys this rule: every finite-volume set with finite perimeter has boundary size at least that of an equal-volume ball in constant curvature kappa. For kappa zero, a bounded positive-volume set attains equality exactly when, ignoring zero-volume differences, it is an open region isometric in its induced metric to a round Euclidean ball.

What does that help mathematicians do?

The second manuscript reports that, in any proper CAT(0) space, compactly supported integral n-cycles for n at least two admit compactly supported integral fillings with the sharp Euclidean mass bound. CAT(0) expresses nonpositive curvature through distances; proper means closed bounded sets are compact. Cycles are generalized closed surfaces carrying integer weights. The claim allows arbitrary integer multiplicities and ambient dimension, so researchers could use the Euclidean benchmark to bound filling mass even in nonsmooth spaces.

Are there practical applications?

The immediate value is foundational: the claimed inequalities give optimal benchmarks for boundary size and filling mass in nonpositively curved geometry. The equality classification would also let researchers deduce a region's exact Euclidean geometry from attaining the boundary bound. It identifies equality regions, not the geometry of the entire surrounding manifold.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity

September 23, 2026 42 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Sharp integral fillings in CAT(0) spaces

September 23, 2026 46 pages

Every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026,
  author = {{OpenAI}},
  title = {{Sharp integral fillings in CAT(0) spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-integral-fillings-in-CAT\%280\%29-spaces-September-23-2026/paper.pdf}{OAI:Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/337.md.

Sharp Cartan–Hadamard isoperimetry and rigidity

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the sharp Euclidean filling inequality for every compactly supported integral nn-cycle in a proper CAT(0)\mathrm{CAT}(0) space, for n≥2n\ge2. It constructs a compactly supported integral filling whose mass is at most CnC_n times the boundary mass to the power (n+1)/n(n+1)/n, where CnC_n is the Euclidean isoperimetric coefficient. Integer multiplicities and the ambient dimension are unrestricted.

A second statement proves that this coefficient is optimal by giving, in Euclidean space, a cycle for which every compactly supported integral filling has at least that mass. This yields the selected integral-current form of the Cartan–Hadamard isoperimetric conjecture.

Comparator links

Result Comparator statement
Optimality of the Euclidean filling coefficient FillingCoefficient.lean
Sharp integral fillings in proper CAT(0)\mathrm{CAT}(0) spaces SharpCAT0Filling.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.