Gigli’s characterization of Alexandrov curvature

1 · The essentials

This family of results connects two ways of describing a space whose curvature stays above a chosen lower bound. One describes curvature through the geometry of distances; the other uses conditions that can be tested without assuming the space is smooth. The main manuscript shows that these descriptions agree in every integer dimension at least two, under specific assumptions. The space must be complete and separable, and its measure must be the specified dimension’s Hausdorff measure, with support throughout the space. The second description requires both the stated RCD condition and a lower bound on sectional curvature tested in Gigli’s original global classes. The manuscript presents this as a resolution of Gigli’s characterization conjecture in these dimensions. A Lean-formalized companion establishes a related result about how a function bends along every shortest path; its stated scope is narrower than the full characterization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine checking a landscape either by looking at the shapes of routes across it or by testing how a height reading bends along those routes. The results connect geometric descriptions with such tests, but only under the precise assumptions given—not for an arbitrary landscape or arbitrary readings.

Why it matters

The characterization connects Alexandrov lower curvature bounds with the RCD framework and distributional sectional curvature. According to the main manuscript, it resolves Gigli’s characterization conjecture in dimensions at least two. The companion’s conclusion concerns every minimizing geodesic, making its scope explicitly path-by-path rather than merely a statement about selected paths.

2 · Going further

Fix an integer $n \geq 2$ and a real number $κ$ . The main manuscript shows that a complete separable metric space is an $n$ -dimensional Alexandrov space with curvature at least $κ$ if and only if, equipped with the reference measure $H^{n}$ , it is a full-support $RCD ((n - 1) κ, n)$ space and has distributional sectional curvature at least $κ$ . The latter condition is imposed in Gigli’s original global test classes. The repository explicitly specifies the unreduced RCD condition.

The source identifies this equivalence as Gigli’s characterization conjecture and describes the main manuscript as resolving it in dimensions at least two. Both directions belong to the stated characterization: Alexandrov geometry implies the specified analytic conditions, and those conditions imply Alexandrov geometry. The choice of reference measure, full support, and test classes are parts of the statement, not optional conventions.

The Lean-formalized companion concerns a full-support $RCD (K, N)$ space with $1 < N < \infty$ . 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. This supplies the family’s stated connection between a weak bound on second derivatives and behavior along individual shortest paths. The abstracts do not describe the further argument linking this result to the full curvature characterization, so they do not support a more detailed reconstruction of that argument.

Useful notions

Metric spaces and shortest paths
Completeness and separability
Hausdorff measure and full support
Lower curvature bounds
Weak derivatives and Hessians

What was known before / what is new

Before

The source identifies Gigli’s characterization conjecture as the problem addressed by the main manuscript. Its formulation uses Alexandrov curvature, an RCD condition, and distributional sectional curvature in Gigli’s original global test classes. The supplied abstracts give no further history or account of earlier partial results.

New

The main manuscript shows the full stated equivalence for every integer $n \geq 2$ and every real $κ$ . The companion establishes the passage from a distributional Hessian upper bound to the corresponding inequality along every minimizing geodesic, under its stated assumptions. That companion result has a Lean formalization.

Limits and assumptions

The characterization concerns complete separable metric spaces, integer dimensions $n \geq 2$ , the reference measure $H^{n}$ , full support, the unreduced $RCD ((n - 1) κ, n)$ condition, and distributional sectional curvature in Gigli’s original global test classes. No claim for dimension one or for altered measures or test classes is supplied. The formalized companion assumes full-support $RCD (K, N)$ with $1 < N < \infty$ , a bounded globally Lipschitz function, and a bounded continuous upper bound for its distributional Hessian. The family has Lean verification at this companion-result scope; the source does not state that Lean verifies the entire Alexandrov characterization. No remaining open problem is specified.

Mini glossary

Alexandrov curvature: The geometric notion of a lower curvature bound appearing on one side of the main characterization.
RCD condition: The curvature-and-dimension condition on the measured metric space required on the other side of the characterization.
Full support: The requirement that the reference measure has support throughout the space.
Distributional sectional curvature: The weak formulation of sectional curvature whose lower bound here is tested in Gigli’s original global test classes.
Distributional Hessian: A weak form of a function’s second derivatives, whose upper bound is the hypothesis of the companion result.
Minimizing geodesic: A shortest path between its endpoints.

Keywords : Alexandrov curvature · RCD spaces · Distributional Hessian

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/356.md

Manuscripts in this family

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Gigli’s distributional curvature characterization of Alexandrov spaces

24/09/2026

To be verified

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 $H^{n}$ , it is a full-support $RCD ((n - 1) κ, n)$ 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.

PDF GitHub folder

Weak Hessian bounds along every geodesic in RCD spaces

24/09/2026

Verified by Lean

On a full-support $RCD (K, N)$ space with $1 < N < \infty$ , 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.

PDF GitHub folder Lean formalization