CATALOGUE 356
Gigli’s characterization of Alexandrov curvature
The two papers establish the equivalence between Gigli’s distributional curvature lower bound and an Alexandrov lower curvature bound, together with the weak-Hessian theorem used in the converse.
The two papers in official catalogue order
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Gigli’s distributional curvature characterization of Alexandrov spaces →Short name: Distributional curvature characterizationSeptember 24, 2026 · 53 pagesOriginal paper (GitHub) ↗ | For integer n≥2 and κ∈ℝ, characterizes n-dimensional CBB(κ) by full-support unreduced RCD((n−1)κ,n) with the exact measure ℋⁿ and Gigli’s distributional curvature lower bound on global TestV. Theorem 1.1 · p. 2 ↗ | The forward implication uses integrated short-flow expansions and local error removal. The converse passes through a heat index inequality, noncollapsed frames and weak distance-Hessian comparison, then uses the companion theorem to obtain triangle comparison. Read the proof overview → |
| 02 | Weak Hessian bounds along every geodesic in RCD spaces →Short name: Weak Hessian boundsSeptember 24, 2026 · 11 pagesOriginal paper (GitHub) ↗ | On a full-support RCD(K,N) space, 1<N<∞, a weak Hessian upper bound G for bounded globally Lipschitz F, with G bounded and continuous, yields (F∘σ)″≤ℓ²G∘σ in distributions along every constant-speed minimizing geodesic σ. Theorem 1.1 · p. 2 ↗ | Bounded measure changes and generator-domain closure yield variable-curvature CD; nonbranching and concentration of endpoint measures localize it to a prescribed geodesic. Direct input to Theorem 6.1 of the main paper, allowing collapse and requiring neither F∈L² nor an L² Hessian. Read the proof overview → |
References on the arrows
- OpenAI26a: OpenAI, Gigli’s distributional curvature characterization of Alexandrov spaces, September 24, 2026. Both directions of the characterization and the application of the weak-Hessian input. [OpenAI26a, Theorem 1.1, p. 2]
- OpenAI26b: OpenAI, Weak Hessian bounds along every geodesic in RCD spaces, September 24, 2026. Passage from a weak Hessian bound to a distributional inequality along every geodesic. [OpenAI26b, Theorem 1.1, p. 2]
Both directions and the companion input
The forward implication first obtains the RCD structure from $\operatorname{CBB}(\kappa)$. Integrated short-flow expansions, local removal of a square error, approximation and exhaustion then recover the curvature inequality on the original global $\operatorname{Test}/\operatorname{TestV}$ classes. The converse builds a heat-regularized index inequality from the curvature lower bound. Frames defined at every time along almost every path of suitable test plans, together with countable trials and noncollapse, turn it into weak Hessian comparison for squared distance. The characterization article explains these steps.
The second paper enters after constructing a bounded global distance modification. Theorem 6.1 of the main paper is precisely Theorem 1.1 of the companion, supplying the model ODE along every minimizing geodesic. Comparison for short triangles yields the local four-point condition and then globalization; the exact measure $\mathcal H^n$ recovers dimension $n$.
[OpenAI26a, Theorem 6.1, p. 48]
[OpenAI26a, Theorem 6.3, pp. 49–50]
The companion itself applies to general full-support finite-dimensional RCD spaces, including collapsed spaces. It requires neither global $L^2$ integrability of $F$ nor an $L^2$ Hessian. The weak-Hessian article explains bounded measure changes, closure in the generator domain and concentration of endpoint measures. Its proof does not depend on the characterization or its frame construction.
Sources and checking scope
The official subject is Differential geometry (333–361). The catalogue and both manuscripts are fixed at commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a; both manuscript versions are dated September 24, 2026.
[OpenAICatalog, Catalogue 356, p. 38]
We compared the principal statements, the core reductions in both directions of the main paper, the companion’s proof text and its application in the main paper against the sources. Each article records the statements and uses of external inputs and the remaining unchecked scope. This does not independently verify every estimate or lemma of the main paper or the complete proofs of external references.