Stability of local Riemannian Ricci curvature lower bounds

Federico Renzi; Federico Vitillaro; Francesco Nobili; Luigi Ambrosio

Stability of local Riemannian Ricci curvature lower bounds

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Pith's one-line read Local Riemannian Ricci lower bounds are stable under Gromov–Hausdorff convergence, and almost every weak tangent is Euclidean.

desk verdict Solid local stability package for RCD that answers Honda–Sun and supplies the missing local EVI/Mosco tools; the doubling dependence is explicit and not a hidden gap. read the letter →

arxiv 2607.27008 v1 pith:OMFPKBPC submitted 2026-07-29 math.DG math.FAmath.MG

Luigi Ambrosio , Francesco Nobili , Federico Renzi , Federico Vitillaro This is my paper

classification math.DGmath.FAmath.MG MSC 53C2349Q2258J35

keywords localRCDGromov-HausdorffstabilityinfinitesimalHilbertianityparallelogramidentityEvolutionVariationalInequalityweaktangentsessentialnon-branchingCheegerenergy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Global synthetic Ricci lower bounds of Riemannian type (RCD) are known to pass to Gromov–Hausdorff limits, but the theory is intrinsically global while many geometric questions are local. This paper defines local versions of those bounds on open sets, with curvature and dimension allowed to vary from point to point, and proves they remain stable under pointed measured Gromov convergence when the local radii and curvature–dimension parameters stay uniformly controlled. The hard step is showing that the parallelogram identity for weak gradients—the infinitesimal Hilbertianity that distinguishes Riemannian from Finsler geometry—survives in the limit. The authors obtain this by a Lagrangian argument that needs a new local Evolution Variational Inequality for the heat flow on small balls, with an error controlled by the decay of the flow. Once stability is secured, weak tangents of spaces with local RCD bounds are themselves global RCD(0,N) spaces, and therefore Euclidean at almost every point.

What carries the argument

An effective local Evolution Variational Inequality for the heat flow on sufficiently small balls, carrying a remainder that vanishes as t→0 when the measure is locally doubling (so the balls are John domains with Gaussian heat-kernel bounds). This yields strong local displacement convexity of entropy and local essential non-branching, which close the Lagrangian Mosco-convergence argument for the parallelogram identity.

What would settle it

Exhibit a pmG-convergent sequence of locally RCD spaces whose local radii and curvature parameters stay controlled, yet the limit fails the parallelogram identity for some pair of compactly supported Sobolev functions, or fails to have an Euclidean weak tangent at a positive-measure set of points.

Watch

Extended reading notes

Core claim

If a sequence of pointed complete metric measure spaces converges in the pointed measured Gromov sense and open sets Ω_n in them satisfy the local RCD condition with curvature–dimension parameters and local radii that remain controlled at every limit point, then the limit open set satisfies the corresponding local RCD condition. When the local radius becomes infinite at even one point, the whole limit space is globally RCD.

Load-bearing premise

The error term in the local heat-flow inequality must vanish for small times, which needs the reference measure to be locally doubling so that small balls admit Gaussian upper bounds on the heat kernel.

Editorial extensions

If this is right

Local RCD bounds pass to Gromov–Hausdorff limits under uniform control of radii and curvature–dimension data.
Weak tangents of locally RCD spaces are globally RCD(0,N) and Euclidean almost everywhere.
Strong displacement convexity of entropy and essential non-branching hold on sufficiently small balls.
The same stability holds in the infinite-dimensional local RCD(K(·),∞) setting once local doubling is assumed.
If the local radius is infinite at one point, the entire limit space is globally RCD.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The local EVI-with-error may be useful for quantitative stability or for Ricci flows that remain only locally controlled.
The John-domain heat-kernel analysis suggests the result extends to other domains (e.g. uniform domains) that support global Sobolev inequalities without full geodesic completeness.
Almost-everywhere Euclidean tangents under purely local curvature bounds open a route to partial regularity for singular spaces that are only locally RCD.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is the local stability theory for RCD that the field has needed. Theorem 1.1 gives pmG-stability of RCD_loc(K(·),N(·)) under uniform control on the local radii and the curvature-dimension parameters, and if the limiting radius is infinite at one point you recover a global RCD space. The tangent application (Thm 1.2) then yields a.e. Euclidean weak tangents under purely local assumptions, via Gigli–Mondino–Rajala. That answers Honda–Sun Question 7.5 and is the main reason to care.

What is genuinely new is the local package that makes the Lagrangian Mosco argument of their earlier arXiv:2511.13320 work in the presence of (possibly negative) curvature: the local EVI with an explicit remainder (Thm 1.4/7.1), the vanishing of that remainder on small John domains once local doubling is in hand (Prop 7.4 + heat-kernel bounds of §6), the resulting strong CD_loc and local essential non-branching (Thm 7.5, Cor 7.6), and the polygonal-plan liminf for Cheeger energy (Thm 8.2). Local CD stability itself (Thms 3.8/3.14) is the expected localization of Sturm/Lott–Villani/GMS and is cleanly done. The writing is full and the logical chain is standard deductive RCD technology; self-citations are tools, not the target.

The soft spot is exactly the one the authors flag: without local doubling the EVI error need not vanish, strong CD_loc fails, and the Mosco argument does not close. That is why the main theorems are stated under finite-dimensional RCD_loc (via Cor 3.12). It is a legitimate hypothesis, not a concealed gap, and the infinite-dimensional statements are carefully restricted. Minor residual risk of a local estimate slip remains, as with any long technical paper, but nothing load-bearing looks broken.

This is for people who work with synthetic Ricci bounds, Ricci limits, or local-to-global questions in metric measure geometry. It deserves a serious referee. I would accept it for peer review and expect it to be used.

Referee Report

0 major / 4 minor

Summary. The paper introduces local curvature-dimension conditions CD_loc(K(·),N(·)) and RCD_loc(K(·),N(·)) on open subsets of metric measure spaces, with point-dependent curvature, dimension and local radius. It proves that these conditions are stable under pointed measured Gromov convergence when the local radii stay uniformly positive and the curvature/dimension parameters remain controlled (Theorems 1.1, 3.8, 3.14). The key analytic step is a local Mosco convergence of Cheeger energies (Theorem 1.3 / Theorem 8.2), obtained by adapting the Lagrangian polygonal approximation of [70] after establishing a local Evolution Variational Inequality with an annular remainder (Theorem 1.4 / Theorem 7.1). The remainder is shown to vanish on sufficiently small balls once local doubling makes them John domains admitting Gaussian heat-kernel bounds (Proposition 7.4, Theorem 6.17). This yields strong CD_loc and local essential non-branching (Theorem 7.5, Corollary 7.6). As an application, almost every point of an RCD_loc space admits a Euclidean weak tangent (Theorem 1.2).

Significance. Stability of global RCD under pmG convergence is classical (Ambrosio–Gigli–Savaré, Gigli–Mondino–Savaré). Extending it to genuinely local, pointwise-variable bounds closes a gap left open by Honda–Sun and aligns the synthetic theory with the extrinsic, local character of Cheeger–Colding. The local EVI with controlled remainder and the resulting strong displacement convexity / essential non-branching on small balls are of independent interest and supply tools for further local analysis. The Euclidean-tangent application via Gigli–Mondino–Rajala is a clean payoff. The arguments are written in full, rely on standard optimal-transport and Dirichlet-form ingredients, and make the doubling hypothesis needed for error vanishing completely explicit.

minor comments (4)

[Theorem 1.1 / §3.2–§8] In the statement of Theorem 1.1 the universal constant c appearing in r_∞(x)=c·lim r_n(y_n) is left unspecified; a parenthetical reference to the concrete factors 1/10 (Theorem 3.14) and 1/6 (Theorem 8.2) would help the reader track the radius losses.
[§1, §7–§8] The reduction from RCD_loc(K(·),N(·)) to strong CD_loc(K(·),∞) via Corollary 3.12 and Theorem 7.5 is correct but slightly buried; a short remark in the introduction or at the beginning of §8 recalling that finite dimensionality supplies the doubling needed for Proposition 7.4 would improve readability.
[Throughout] Typographical inconsistencies appear in a few places (e.g., “comple metric”, “Poincar´ e”, “Arzel` a-Ascoli”). A light copy-edit pass would remove them.
[Definition 3.1, Remark 8.3] In Definition 3.1 the requirement r(x)≤r_C(x) is natural for locally complete spaces, but the subsequent global statements sometimes silently assume completeness of the ambient space; Remark 8.3 clarifies the issue, yet a forward pointer earlier would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: local RCD stability is proved from OT/Dirichlet-form ingredients, not by renaming inputs

full rationale

The paper defines CD_loc/RCD_loc synthetically, then proves stability under pmG by (i) localizing classical entropy-convexity stability, (ii) establishing a local EVI with an explicit remainder, (iii) killing the remainder under local doubling via John-domain Gaussian heat-kernel bounds from the Dirichlet-form literature, yielding strong CD_loc and local essential nonbranching, and (iv) feeding those into a Lagrangian polygonal Mosco argument adapted from the authors' prior work [70]. Self-citations ([70], AGS, etc.) supply tools and comparison theorems; none of the target identities (parallelogram stability, RCD_loc stability, Euclidean weak tangents) is assumed in those citations or forced by a fit/normalization. There are no fitted constants, no uniqueness theorem that merely restates the claim, and no step where a predicted quantity equals an input by construction. The derivation is self-contained mathematical deduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper works inside the standard synthetic Ricci / optimal-transport axiomatics. No free parameters are fitted. Load-bearing background is the global CD/RCD calculus, heat-kernel Gaussian bounds under doubling+Poincaré/Nash, and the Lagrangian Mosco scheme; local definitions and the error-controlled EVI are the paper's own contributions, not new physical entities.

assumptions (6)

domain assumption Metric measure spaces are locally complete, separable, with boundedly finite measures satisfying the quadratic exponential growth condition (2.1) when unbounded.
Standing setup in §2.1; needed for Wasserstein space, entropy, and heat flow to be well-defined.
domain assumption CD(K,N)/CD(K,∞) entropy convexity along some (resp. all, for strong CD) W2-geodesics characterizes synthetic Ricci lower bounds (Lott–Villani, Sturm).
Definitions 3.1 and 3.9 localize this standard axiom; stability proofs reduce to it.
domain assumption Infinitesimal Hilbertianity is equivalent to the parallelogram identity for weak gradients on functions supported in the open set (Thm 2.14 / Gigli).
Used to pass from Mosco convergence of Cheeger energies to RCD_loc in Thm 1.1 and 8.1.
standard math On doubling John domains supporting a weak (2,2)-Poincaré inequality, the Dirichlet form of a quadratic Cheeger energy admits a Hölder heat kernel with Gaussian upper bounds (Sturm; Chen–Kim–Kumagai–Wang; Hajłasz–Koskela Sobolev on John domains).
Invoked in §6 (Thm 6.8, 6.16, 6.17) to kill the local-EVI error term.
domain assumption Under strong CD_loc, absolutely continuous marginals on small balls admit a unique nonbranching optimal geodesic plan induced by a map (Rajala–Sturm type, Thm 4.3).
Corollary 7.6; required for the polygonal approximation controlling compression in Thm 8.2.
domain assumption pmG convergence is realized extrinsically by isometric embeddings into a common space Z with weak convergence of measures (Gigli–Mondino–Savaré).
Definition 2.20; all stability statements are relative to this topology.

invented entities (2)

CD_loc(K(·),N(·)) / RCD_loc(K(·),N(·)) with pointwise local radius r(x) independent evidence
purpose: Encode Ricci/dimension bounds only on small balls with parameters allowed to vary in space.
Definitions 3.1, 3.9, 3.15–3.16; standard localization of CD/RCD, not a new physical object. Independent checks are the stability and tangent theorems themselves.
Local EVI with remainder C·lim∥f_t∥_L∞(annulus) along the localized heat flow independent evidence
purpose: Replace the global EVI so that strong displacement convexity and essential nonbranching can be recovered on small balls.
Theorem 1.4 / §7.1; the remainder is an analytic error term, not a new geometric entity. Falsifiable inside the theory by whether the error vanishes under doubling.

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Pith. "Pith review of Stability of local Riemannian Ricci curvature lower bounds." pith.science (2026). https://pith.science/paper/OMFPKBPC

@misc{pith2026260727008,
  author       = {Pith},
  title        = {Pith review of: Stability of local Riemannian Ricci curvature lower bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMFPKBPC}},
  note         = {Machine review of arXiv:2607.27008}
}

read the original abstract

We establish the stability of local Riemannian Ricci curvature lower bounds along Gromov-Hausdorff convergence. A central part of our analysis is devoted to showing the stability of the parallelogram identity for weak gradients, obtained implementing the Lagrangian approach developed in [arXiv:2511.13320] in the local setting. As an application, we deduce the almost everywhere existence of Euclidean weak tangents. An important ingredient, of independent interest, is an effective local Evolution Variational Inequality along the heat flow on sufficiently small balls, with a remainder term depending on the rate of decay of the flow. This has applications to strong displacement convexity of the Entropy functional along local Wasserstein interpolations and to local essential nonbranching properties.

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Reviewed July 30, 2026 · model on record in the stance chip above.

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Cavalletti and E

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