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arxiv: 2606.11085 · v2 · pith:TTKXK7OBnew · submitted 2026-06-09 · 🧮 math.PR · math.MG· math.SP

Geometric obstructions to Lipschitz transport between weighted Hessian CD(kappa,infty) manifolds

William Dudarov , Dan Mikulincer This is my paper

Pith reviewed 2026-06-27 11:46 UTC · model grok-4.3

classification 🧮 math.PR math.MGmath.SP
keywords weighted Riemannian manifoldscurvature-dimension conditionLipschitz transportGaussian measuresweighted LaplacianWeyl asymptoticstransport maps
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The pith

There exists a weighted manifold on R² satisfying CD(1/2,∞) with no Lipschitz map from Euclidean space pushing the centered Gaussian to its measure.

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

The paper constructs a weighted Riemannian manifold on two-dimensional space that obeys the CD(1/2,∞) curvature-dimension condition. For this manifold the centered Gaussian on flat R² cannot be pushed forward by any Lipschitz map to the manifold's weighted measure. The authors derive a Weyl-type asymptotic for the eigenvalues of the associated weighted Laplacian and establish that these eigenvalues grow much more slowly than the eigenvalues of the Gaussian Laplacian. The construction supplies counterexamples showing that the CD condition alone does not force the existence of Lipschitz transports between the spaces.

Core claim

The central claim is the construction of a weighted Riemannian manifold (R²,g,μ) satisfying CD(1/2,∞) such that no Lipschitz map T:(R²,‖·‖)→(R²,g) exists with T#γ=μ, where γ is the centered Gaussian on R². This is supplemented by a Weyl-type asymptotic law for the eigenvalues of −Δ_{g,μ} showing they are asymptotically negligible compared with the eigenvalues of −Δ_γ.

What carries the argument

The weighted Riemannian manifold (R²,g,μ) satisfying the CD(1/2,∞) condition, which is engineered to block any Lipschitz transport of the centered Gaussian measure.

If this is right

  • The eigenvalues of the weighted Laplacian −Δ_{g,μ} obey a Weyl-type asymptotic law.
  • These eigenvalues are asymptotically negligible relative to the eigenvalues of −Δ_γ.
  • The manifold serves as a counterexample showing that the CD condition does not guarantee Lipschitz transport from Euclidean space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric obstruction could be tested in higher dimensions to see whether CD bounds continue to permit or block Lipschitz maps.
  • One could examine whether the eigenvalue comparison extends to other measures or to non-Lipschitz classes of maps.

Load-bearing premise

A weighted manifold on R² can be built that satisfies the CD(1/2,∞) condition while admitting no Lipschitz map from Euclidean space that pushes the centered Gaussian to its measure.

What would settle it

Explicitly exhibiting a Lipschitz map T from (R² with the Euclidean norm) to the constructed (R²,g) such that T#γ=μ would disprove the claimed obstruction.

Figures

Figures reproduced from arXiv: 2606.11085 by Dan Mikulincer, William Dudarov.

Figure 1
Figure 1. Figure 1: Moment measures (R d , ∇2W, µ) µ = e −W(x) dx (R d , ∇2W∗ , ν) ν = (∇W)#µ = e −V (x) dx (R d , ∥ · ∥, γ) γ = 1 (2π) d/2 e −∥x∥ 2/2 dx T S ∇W . transport map from ν to µ. We equip the two spaces with the Hessian metrics induced by this pair of convex functions (R d , ∇2W, µ) and (R d , ∇2W∗ , ν). Differentiating the identity ∇W(∇W∗ (x)) = x gives ∇2W(∇W∗ (x)) ∇2W∗ (x) = Id, or equivalently ∇2W∗ (x) = ∇2W(∇W… view at source ↗
read the original abstract

We construct a weighted Riemannian manifold $(\mathbb R^2,g,\mu)$ satisfying $\mathrm{CD}(1/2,\infty)$, the curvature-dimension condition, with the following property: if $\gamma$ denotes a centered Gaussian measure on $\mathbb R^2$, then there is no Lipschitz map $T:(\mathbb R^2,\|\cdot\|) \to (\mathbb R^2,g)$ satisfying $T_\#\gamma=\mu$. Building on this, we prove a Weyl-type asymptotic law for the eigenvalues of the weighted Laplacian $-\Delta_{g,\mu}$ and show that they are asymptotically negligible when compared to the eigenvalues of $-\Delta_{\gamma}$. These results give strong counterexamples to two questions of E. Milman and complement the recent counterexample of Aryan.

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

0 major / 3 minor

Summary. The manuscript constructs an explicit weighted Riemannian manifold (ℝ², g, μ) satisfying the CD(1/2, ∞) curvature-dimension condition for which no Lipschitz map T:(ℝ², ‖·‖) → (ℝ², g) exists with T#γ = μ, where γ is the centered Gaussian on Euclidean ℝ². It further derives a Weyl-type asymptotic for the eigenvalues of the weighted Laplacian −Δ_{g,μ} showing they are asymptotically negligible relative to those of −Δ_γ. These results furnish counterexamples to two questions posed by E. Milman and complement the recent counterexample of Aryan.

Significance. If the explicit construction and curvature verification hold, the paper supplies a concrete geometric obstruction separating the CD(κ, ∞) condition from the existence of Lipschitz transport from Gaussian space. The eigenvalue comparison strengthens the result by exhibiting a spectral consequence. This advances the understanding of when curvature-dimension conditions control transport and functional inequalities on weighted manifolds.

minor comments (3)
  1. [§1] §1 (Introduction): the statement that the construction 'complements the recent counterexample of Aryan' should include a precise citation to Aryan's work for immediate context.
  2. [§2] The definition of the metric g and the density of μ (presumably in §2 or §3) should be stated with explicit coordinate formulas before the verification of CD(1/2, ∞) begins, to allow direct checking of the curvature bound.
  3. [§4] In the eigenvalue asymptotic (likely §4), clarify whether the Weyl law is proved via the standard Minakshisundaram–Pleijel expansion or via a different comparison argument; the current phrasing leaves the method implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on the CD(1/2, ∞) counterexample to Lipschitz transport and the Weyl-type eigenvalue asymptotics. The recommendation is for minor revision, but the report contains no specific major comments requiring a response. We therefore make no changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; explicit construction is self-contained

full rationale

The paper's central result is an explicit construction of a weighted manifold (R²,g,μ) satisfying CD(1/2,∞) while obstructing Lipschitz transport from the Gaussian γ. The abstract and skeptic analysis present this construction, along with consequent Weyl asymptotics for the weighted Laplacian, as the load-bearing step. No equations or claims reduce by definition to their own outputs, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The derivation chain is independent of the target result and stands on the details of g and μ supplied in the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no details on free parameters, axioms, or invented entities are visible.

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Reference graph

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