Result 356, Differential geometry

Gigli’s characterization of Alexandrov curvature

Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support RCD((n−1)κ,n)\mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n) condition with reference measure Hn\mathcal H^n and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.

Lean formalization Proof

The bigger picture

Why it matters

Even spaces with singularities can have curvature bounds expressed through comparisons of triangles. This manuscript claims an exact match between that geometric viewpoint and analytic curvature conditions, giving researchers two equivalent ways to recognize the same spaces.

What changes?

For complete separable metric spaces, the manuscript reports an equivalence in every integer dimension n at least two and for every real kappa. Alexandrov curvature at least kappa in dimension n is equivalent to the unreduced RCD condition, with Ricci bound (n minus one) times kappa and dimension parameter n, plus distributional sectional curvature at least kappa in Gigli's original global test classes. The reference measure is n-dimensional Hausdorff measure, with full support: every nonempty open set has positive measure.

What does that help mathematicians do?

RCD expresses a weak, Riemannian version of a lower Ricci-curvature bound; distributional sectional curvature adds finer information through tests against functions. The claimed equivalence means that, under the stated assumptions, these analytic constraints force geometric triangle comparisons, even without a smooth metric. Conversely, Alexandrov geometry supplies those analytic bounds. This lets researchers move between geometric comparisons and calculus formulated without ordinary derivatives.

Are there practical applications?

Its immediate value is foundational: it identifies exactly when this analytic description captures Alexandrov geometry, including nonsmooth spaces. Researchers can use the characterization to check whether proposed examples belong to that class. The claim depends on both curvature conditions and the specified measure; it does not make the RCD condition alone sufficient.

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

Gigli’s distributional curvature characterization of Alexandrov spaces

September 24, 2026 53 pages

For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure Hn\mathcal H^n, it is a full-support RCD((n−1)κ,n)\mathrm{RCD}((n-1)\kappa,n) space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026,
  author = {{OpenAI}},
  title = {{Gigli's distributional curvature characterization of Alexandrov spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf}{OAI:Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026}},
  year = {2026}
}

Weak Hessian bounds along every geodesic in RCD spaces

September 24, 2026 11 pages Main result formalized in Lean

On a full-support RCD(K,N)\mathrm{RCD}(K,N) space with 1<N<∞1\lt N\lt \infty, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026,
  author = {{OpenAI}},
  title = {{Weak Hessian bounds along every geodesic in RCD spaces}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf}{OAI:Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/356.md.

Gigli’s characterization of Alexandrov curvature

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

Scope

The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an RCD(K,N)\mathrm{RCD}(K,N) space with finite N>1N>1. If a bounded globally Lipschitz function FF has weak Hessian bounded above by GG times the metric, where GG is bounded and continuous, then every constant-speed geodesic γ:[0,1]→X\gamma:[0,1]\to X satisfies (F∘γ)′′≤G(γ) d(γ(0),γ(1))2(F\circ\gamma)''\le G(\gamma)\,d(\gamma(0),\gamma(1))^2 in the distributional sense. Constant geodesics are included. The space is complete and separable with full support and measure finite on bounded sets; compactness and metric nonbranching are not assumed.

Comparator links

Result Comparator statement
Weak Hessian bounds along every geodesic WeakHessian.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.