Browse › Differential geometry › 356 Gigli’s characterization of Alexandrov curvature
Gigli’s distributional curvature characterization of Alexandrov spaces
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
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 note for family 356 (lean/docs/356.md, 2026). The release's Lean development covers only the companion theorem on weak Hessian bounds along every geodesic, which this paper uses in the converse direction; the forward direction, the heat-regularized index inequality and the frame constructions of this paper are not covered. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Alexander Lytchak, Stephan Stadler, Ricci curvature in dimension 2 (arXiv:1812.08225; J. Eur. Math. Soc. 25 (2023), 845–867). Proves that if (X,d,H^2) is RCD(κ,2) then X is Alexandrov with curvature at least κ, so in the case n=2 of Theorem 1.1 the implication (B)⇒(A) follows without the distributional curvature hypothesis; it is not in the paper's bibliography.
Digests and expositions
- Latent Seminar, Alexandrov curvature: from distributional tensors to every geodesic (7 October 2026). This exposition outlines both manuscripts’ proof mechanisms, including exponential reweighting, concentration onto prescribed geodesics, and the index-to-Hessian comparison argument, while explicitly disclaiming independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 356
Independent sites; their content is not reviewed here.
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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/356.md. The Lean statements have not been checked by people against the paper's claims.
Family 356: 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.
Other papers in this family:
- Weak Hessian bounds along every geodesic in RCD spaces · September 24, 2026