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arxiv: 2509.14209 · v2 · submitted 2025-09-17 · 🧮 math.MG · math.DS· math.PR

Characterization of foliations via disintegration maps

Florentin M\"unch , Renata Possobon , Christian S. Rodrigues This is my paper

Pith reviewed 2026-05-18 15:50 UTC · model grok-4.3

classification 🧮 math.MG math.DSmath.PR
keywords disintegration mapsmetric measure foliationsWasserstein spaceconditional measuresfoliation characterizationoptimal transportgeometric arrangement
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The pith

Disintegration maps characterize which conditional measures arise from metric measure foliations.

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

The paper develops a method to connect the supports of conditional measures with their geometric arrangement in Wasserstein space through the disintegration map. It sets out criteria that identify when those measures come from a metric measure foliation rather than some other arrangement. A reader would care because the approach gives a concrete way to study foliations inside optimal transport spaces. The authors illustrate the method with an example on perturbations of disintegration-induced foliations.

Core claim

The central claim is that the disintegration map encodes the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space, and this encoding supplies criteria that determine precisely when the conditional measures arise from a metric measure foliation.

What carries the argument

The disintegration map, which links supports of conditional measures to their positions in Wasserstein space and thereby distinguishes foliated from non-foliated arrangements.

Load-bearing premise

The disintegration map carries enough geometric information about the supports to separate foliated arrangements from non-foliated ones.

What would settle it

A collection of conditional measures whose disintegration map satisfies the stated criteria yet whose supports do not form a metric measure foliation would falsify the characterization.

Figures

Figures reproduced from arXiv: 2509.14209 by Christian S. Rodrigues, Florentin M\"unch, Renata Possobon.

Figure 1
Figure 1. Figure 1: Disintegration of Leb2 with respect to Leb1 : the red line {x} × [0, 1] carries µx = Leb1 In this case, the disintegration map is given by f : [0, 1] → (Pp([0, 1] × [0, 1]), Wp) x 7→ µx, where µx is Leb1 on {x} × [0, 1]. This is a basic example that fits into the broader framework of metric foliations, which is the primary focus of our study. Let (X, d) be a metric space, and F a partition of X into closed… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration for the case in which the measures µy do not have full support: there is isometry between Wp and dY , but {π −1 (y)} (the ellipses) does not define a metric foliation. Note that in the previous example, if λ = 1, that is, the preimages are circles in X, then {π −1 (y)} forms a metric foliation. In the context of Theorem A, we aim to demonstrate that the p-energy exhibits sensitivity to small p… view at source ↗
Figure 3
Figure 3. Figure 3: Normalized arc length as a function of θ for λ equals to 1, 1.1, 1.5 and 2. and Ry ′(θ) = y ′ p λ2 − (λ2 − 1) cos2(θ) . Then, W1(µy, µy ′) = Z Ry ′(θ) − Ry(θ) dµy = Z y ′ − y p λ2 − (λ2 − 1) cos2(θ) y p 1 − (1 − λ−2) cos2(θ) Ly δy dθ = (y ′ − y) y λ Ly Z 2π 0 dθ = 2π (y ′ − y) y λ Ly . Therefore, considering ˆfλ the disintegration map of µ with respect to ν, we have E1( ˆfλ) = 2πy Ly , that is, the energy … view at source ↗
Figure 4
Figure 4. Figure 4: depicts the dependence of E1( ˆfλ) on λ. As shown, E1 grows smoothly with increasing foliation perturbation [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

In this paper, we present a novel approach for analyzing the relationship between the supports of conditional measures and their geometric arrangement in Wasserstein space via the disintegration map. Our method establishes criteria to determine when such conditional measures arise from a metric measure foliation. Additionally, we provide a example demonstrating how this framework can be applied to study perturbations of disintegration-induced foliations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework that uses disintegration maps to relate the supports of conditional measures to their geometric arrangement in Wasserstein space. It claims to establish criteria determining when such conditional measures arise from a metric measure foliation and supplies a perturbation example to illustrate the approach.

Significance. If the claimed criteria can be rigorously derived from the disintegration map and the perturbation example is worked out in detail, the work would supply a new characterization tool at the interface of optimal transport and geometric measure theory. It would be of interest to researchers studying foliations and disintegrations of measures, provided the arguments are made fully explicit.

major comments (2)
  1. [Abstract and introduction] The manuscript states that criteria are established but contains no theorem statement, no explicit definition of the disintegration map in the present context, and no derivation showing how the map distinguishes foliated from non-foliated arrangements. This absence is load-bearing for the central claim.
  2. [Example section] The perturbation example is mentioned but not developed; no concrete construction, no computation of the relevant disintegration maps, and no verification that the perturbed object satisfies or violates the claimed criteria appear in the text. Without these details the example cannot support the general statement.
minor comments (2)
  1. [Abstract] The phrase 'a example' should be corrected to 'an example'.
  2. [Introduction] Notation for the disintegration map and the Wasserstein space is introduced without prior definition or reference to standard sources (e.g., Villani or Ambrosio–Gigli–Savaré).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that greater explicitness in the statement of the main criteria and fuller development of the example will improve the paper. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract and introduction] The manuscript states that criteria are established but contains no theorem statement, no explicit definition of the disintegration map in the present context, and no derivation showing how the map distinguishes foliated from non-foliated arrangements. This absence is load-bearing for the central claim.

    Authors: We acknowledge that the abstract and introduction describe the intended criteria but do not contain a numbered theorem or a self-contained derivation. In the revised version we will add a formal theorem that (i) recalls the disintegration map adapted to the Wasserstein setting, (ii) states the precise criteria under which the supports of the conditional measures arise from a metric-measure foliation, and (iii) sketches the argument showing how the map separates foliated from non-foliated configurations. revision: yes

  2. Referee: [Example section] The perturbation example is mentioned but not developed; no concrete construction, no computation of the relevant disintegration maps, and no verification that the perturbed object satisfies or violates the claimed criteria appear in the text. Without these details the example cannot support the general statement.

    Authors: We agree that the perturbation example is only sketched. The revision will supply an explicit construction of the perturbed measure, compute the associated disintegration maps in both the original and perturbed cases, and verify whether the criteria are preserved or violated, thereby illustrating the framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and high-level description contain no equations, derivations, or explicit self-citations that reduce claims to inputs by construction. The method is presented as establishing criteria for conditional measures arising from metric measure foliations using disintegration maps in Wasserstein space, building on standard disintegration theory and optimal transport without visible self-definitional loops, fitted predictions renamed as results, or load-bearing self-citations. The central claim remains independent of its own definitions, consistent with external benchmarks in the field, yielding a self-contained framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

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Reference graph

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