Browse › Differential geometry › 356 Gigli’s characterization of Alexandrov curvature
Weak Hessian bounds along every geodesic in RCD 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
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.
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 note describes a Lean formalization of Theorem 1.1 of this paper: weak Hessian upper bounds transfer to every constant-speed geodesic in RCD(K,N) spaces, 1<N<∞, and the comparator statement is in
WeakHessian.lean. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Nicola Gigli, Christian Ketterer, Kazumasa Kuwada, Shin-ichi Ohta, Rigidity for the spectral gap on RCD(K,∞)-spaces (arXiv:1709.04017, 2017). Their Theorem 3.3 ("Hessian bound implies convexity") shows that a bounded v with Hess v ≥ −κ gives (−κ)-convexity of μ ↦ ∫v dμ along W2-geodesics between absolutely continuous measures, an antecedent of the paper's Hessian-to-geodesic step under stronger regularity; 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
- Emergent Mind: family 356
Independent sites; their content is not reviewed here.
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}
}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:
- Gigli’s distributional curvature characterization of Alexandrov spaces · September 24, 2026