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Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures

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Pith's one-line read The Martinet structure fails the measure contraction property for every K and N.

desk verdict Solid Martinet/Engel counterexample plus a new quotient toolbox; the sweeping generic conclusions rest on an open conjecture, so read the conditional statements carefully. read the letter →

arxiv 2505.09681 v2 pith:ONDVGJMI submitted 2025-05-14 math.DG math.MGmath.OC

Samuël Borza , Luca Rizzi This is my paper · ORCID
classification math.DGmath.MGmath.OC MSC 53C1753C2349Q22
keywords measurecontractionpropertysub-RiemanniangeometryCarnotgroupsMartinetstructuresyntheticRiccicurvaturequotientsbyisometricgroupactionsessentialnon-branchingminimizingSard
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

This paper establishes that the measure contraction property (MCP), a synthetic lower bound on Ricci curvature phrased in optimal-transport terms, can fail in sub-Riemannian geometry once the step is at least three. The failure already occurs in the Martinet structure, the simplest non-Lipschitz Carnot homogeneous space, and it propagates to the Engel group and beyond. The proof introduces a new stability result: the local MCP descends to quotients by isometric group actions under a weakened non-branching condition that follows from the open minimizing Sard conjecture. If the paper is right, then synthetic Ricci lower bounds of MCP type are not a universal feature of sub-Riemannian and Carnot homogeneous spaces beyond step two, and failure is generic in high dimension.

What carries the argument

The operative mechanism is the delta-essentially non-branching condition: non-branching is demanded only for Wasserstein geodesics between absolutely continuous measures and finite sums of Dirac masses. Unlike the classical essentially non-branching condition, this variant is implied by the star-minimizing Sard property in sub-Riemannian spaces, so it becomes available under a standard conjectural hypothesis. The second piece is a factorization theorem: any quotient between Carnot homogeneous spaces decomposes into finitely many local metric measure isometries and quotients by compact abelian groups, each step preserving the local MCP.

What would settle it

Compute the reduced Jacobian ratio |JR(omega t, m)| / |JR(omega, m)| for a fixed t in (0,1) with m close to 1 and omega large: the paper's expansion shows the double limit is 0, so inequality (51) fails for every finite N, decisively falsifying MCP(0,N) for the Martinet structure. For the propagation results, a concrete counterexample to the minimizing Sard conjecture in a Carnot group of step at most s would invalidate the conditional corollaries.

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Extended reading notes

Core claim

The central discovery is that the Martinet structure does not satisfy MCP(K,N) for any K in the real line and any N in [1, infinity). The argument computes the Jacobian of the sub-Riemannian exponential map from the optimal synthesis of the Martinet flat case and shows that a ratio of reduced Jacobians tends to zero as the elliptic parameter m approaches 1 and the frequency omega tends to infinity, violating the inequality that the MCP would impose. Around this core, the paper builds a quotient calculus: delta-essentially non-branching metric measure spaces preserve the local MCP under discrete and compact abelian isometric quotients, and every quotient between Carnot homogeneous spaces factorizes into such steps. Applying this calculus, the failure propagates upward: any Carnot group admitting a Martinet quotient, including the Engel group and all free step-3 groups, fails every MCP(K,N), conditional on the minimizing Sard property, and generic rank-3 structures in high dimension fail as well.

Load-bearing premise

The propagation of MCP failure from the Martinet structure to Carnot groups and generic structures assumes the minimizing Sard property for all Carnot groups of step at most s, an open conjecture; the Martinet failure itself does not rely on this assumption.

Editorial extensions

If this is right

  • The Engel group, free Carnot groups of step 3, filiform Carnot groups of step at least 3, and rank-2 step-4 Carnot groups fail MCP(K,N) for every K and N; several of these failures are unconditional because the minimizing Sard property is known there.
  • Sub-Riemannian structures with pre-medium-fat distribution can fail the MCP, answering a question raised in the cited literature.
  • Ideal sub-Riemannian structures of rank greater than 3 and sufficiently high dimension generically fail the MCP, even though they have no nontrivial abnormal geodesics.
  • Weaker variants of the MCP, including an entropic version and the quasi curvature-dimension condition QCD, also fail for the same spaces.
  • The Grushin plane satisfies the MCP with sharp Heisenberg constants by a computation-free quotient argument, matching earlier direct estimates.

Reading between the lines

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

  • The paper's dichotomy suggests that a strong Goh-Legendre abnormal geodesic may be the dividing line: local Lipschitz regularity, subanalyticity of the squared distance, and the MCP all fail together, although a direct connection is not yet established.
  • The conditional negative results could be made unconditional by resolving the minimizing Sard conjecture for Carnot groups; preliminary step-4 and step-5 computations mentioned in the paper point in that direction.
  • The quotient technique could be sharpened on the remaining unclassified groups such as N6,2,6 or N6,3,1a: proving MCP failure for either would force failure for several other groups in the low-dimensional table.
  • The same factorization argument appears adaptable to sub-Finsler geometry, where analogous measure-contraction questions remain open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the measure contraction property (MCP) in sub-Riemannian geometry. Its main unconditional result is that the Martinet Carnot homogeneous space fails MCP(K,N) for all K in R and N in [1,∞), proved via the explicit optimal synthesis of Martinet and an asymptotic expansion of the Jacobian of the exponential map. The authors also introduce a weakened non-branching condition, δ-essential non-branching, which is implied by the minimizing Sard property, and prove stability of the local MCP under quotients by discrete and compact abelian isometric group actions. These tools are combined in a factorization argument showing that MCP descends along quotients of Carnot homogeneous spaces, conditional on the minimizing Sard property for Carnot groups. Applications include failure of MCP for Carnot groups admitting a Martinet quotient, with unconditional consequences for step-3, filiform, and rank-2 step-4 groups, and a generic failure statement in high dimension that is conditional on the same open Sard conjecture.

Significance. The unconditional Martinet counterexample is a substantial contribution: it disproves the expectation that MCP holds for all Carnot homogeneous spaces and identifies non-Lipschitzness of the distance as the source of failure. The δ-essentially non-branching condition and the quotient stability theorems are of independent interest and are developed in detail, with the proofs largely self-contained. The paper is honest about the main hypotheses. However, several of the headline applications, including the generic failure theorem, are conditional on the open minimizing Sard conjecture for Carnot groups, and a number of finite verifications in the low-dimensional classification are omitted. The unconditional core is strong enough to merit publication after revision.

major comments (2)
  1. [Section 7.1, Theorem 7.2; Section 8, Theorems 8.1--8.2] The propagation of MCP-failure from Martinet to Carnot groups of step at least 3 and to generic sub-Riemannian structures assumes the minimizing Sard property for all Carnot groups of step at most s (or for all Carnot groups in Theorem 8.1). This is exactly Agrachev's Problem 3 and the Rifford--Trélat Conjecture 1, an open problem. The assumption is stated clearly in the theorems, but the abstract and the introduction present the generic failure as an established fact ('this actually happens generically'). Please reformulate the affected statements as conditional theorems, and ensure that the unconditional content, namely Corollary 7.3 and its precursors, is explicitly separated from the Sard-conditional results.
  2. [Section 7.3, proof of Theorem 7.6 and Table 2; Remark 1.15] The classification of indecomposable Carnot groups of dimension at most 7 is not fully verifiable from the manuscript. The proof states that the subspaces h2,h3 verifying Theorem 6.13 are given in Table 2 and that 'the computations are omitted', and later 'We omit the computations' for the Goh--Legendre checks. Remark 1.15 defers the step-4 and step-5 Sard verifications to 'preliminary computations' not included in the paper. Because Theorem 7.6 is a classification result, these omitted verifications are load-bearing. Please include the computations, either in the text or in an appendix/supplement, or explicitly mark every table entry whose status depends on such omitted verification as conditional.
minor comments (5)
  1. [Section 3.2, proof of Theorem 3.6, Step 3] The second displayed definition in Step 3 repeats the symbol ν_left; it should be ν_right. This makes the mixing construction hard to follow.
  2. [Table 1 and Theorem 7.6] The red/green color coding is essential to the statement; please add symbols or hatching so that the distinction survives non-color printing.
  3. [Remark 6.14] The sufficient condition (38) is quoted from [AG01, Lemma 4]; adding the precise reference inside the proof of Remark 6.14 would help the reader verify the dimension count.
  4. [Corollary 7.3, third bullet] The claim that rank-2 step-4 Carnot groups always admit a quotient to the Martinet structure is asserted as 'easy to check' without any argument; a one-line verification would make the unconditional corollary self-contained.
  5. [Section 9, after equation (47)] The expression 'y(t) = A cn ωt + φ | m' should read 'y(t) = A cn(ωt + φ | m)' for consistency with the Jacobi elliptic function notation used in the surrounding equations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Martinet failure is computed from the external ABCK97 synthesis, and the quotient machinery is proved rather than assumed.

full rationale

The paper's central negative result, Theorem 9.6, is derived from the known optimal synthesis of the Martinet structure [ABCK97] through an explicit Jacobian formula (Proposition 9.3), a necessary inequality imposed by MCP (Proposition 9.5), and a direct two-parameter asymptotic computation (equation 53). The target MCP conclusion is not an input to that calculation. The quotient-transfer results (Theorems 3.5, 4.1, 5.3, 7.1) are proved inside the paper from optimal transport arguments and the new δ-essential non-branching condition; the latter is connected to the minimizing Sard property, which is an explicitly identified open conjecture, not a restatement of the MCP results. The algebraic characterizations (Theorems 6.13 and 6.16) are proved in the text. Self-citations such as [BMR24] and [MR20] provide Sard-related technical lemmas and the fact that branching segments are abnormal; these are independent prior results with stated assumptions, and they do not assume the paper's conclusions. Conditional entries in Table 1 are explicitly marked as depending on the Sard conjecture, which is a limitation of scope rather than a circular step. I find no equation or fitted parameter that is equivalent to the target claim by construction.

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

The central Martinet counterexample is self-contained given the classical optimal synthesis of [ABCK97]; no free parameters are fitted. The quotient machinery depends on the minimizing Sard conjecture to establish delta-essential non-branching of the intermediate factors, which is the main external assumption. No new physical entities are postulated.

assumptions (2)
  • domain assumption Minimizing Sard property holds for all Carnot groups of step <= s (and for all Carnot groups in Theorem 8.1).
    Open conjecture in sub-Riemannian geometry; used via Theorem 3.5 to make intermediate factors delta-essentially non-branching, so that Theorems 7.1, 7.2, 8.1, and 8.2 can propagate MCP failure through quotients.
  • domain assumption Real-analytic sub-Riemannian structures have equivalent minimizing and *-minimizing Sard properties.
    Stated in Section 3.1; needed to apply Theorem 3.5 to real-analytic Carnot homogeneous spaces, which are the spaces treated in the main applications.

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Pith. "Pith review of Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures." pith.science (2026). https://pith.science/paper/ONDVGJMI

@misc{pith2026250509681,
  author       = {Pith},
  title        = {Pith review of: Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ONDVGJMI}},
  note         = {Machine review of arXiv:2505.09681}
}
read the original abstract

We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. \'Ec. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension.

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