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 \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. (Lean)

Manuscripts (2)

Gigli’s distributional curvature characterization of Alexandrov spaces

September 24, 2026 · source · cite as \cite{OAI:Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-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.

Weak Hessian bounds along every geodesic in RCD spaces

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-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.

Result 356 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.