Gigli’s characterization of Alexandrov curvature

2 papersLean

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

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Gigli’s distributional curvature characterization of Alexandrov spaces

OpenAI · September 24, 2026

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 \(\mathcal H^n\), it is a full-support \(\mathrm{RCD}((n-1)\kappa,n)\) 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.

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Cite

@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

OpenAI · September 24, 2026

On a full-support \(\mathrm{RCD}(K,N)\) space with \(1\lt N\lt \infty\), 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.

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Cite

@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}
}

What was formalized in Lean

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

Weak Hessian bounds along every geodesic in RCD spaces

Scope

The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an \(\mathrm{RCD}(K,N)\) space with finite \(N>1\). If a bounded globally Lipschitz function \(F\) has weak Hessian bounded above by \(G\) times the metric, where \(G\) is bounded and continuous, then every constant-speed geodesic \(\gamma:[0,1]\to X\) satisfies \((F\circ\gamma)''\le G(\gamma)\,d(\gamma(0),\gamma(1))^2\) 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

Scope note on GitHub →