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Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition

T0 review · 2 major / 8 minor · reviewed 2026-07-07 · glm-5.2

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Pith's one-line read Near-extremal Sobolev functions stay √ε-close to bubbles

desk verdict Solid paper: first quantitative stability for Sobolev under CD conditions, clean rearrangement strategy, sharp exponent with explicit examples. read the letter →

arxiv 2607.05304 v1 pith:XABY2CWZ submitted 2026-07-06 math.AP math.DGmath.MG

Max Fathi , Ivan Yuri Violo This is my paper · ORCID
classification math.APmath.DGmath.MG MSC 46E3558J0560B05
keywords Sobolevinequalityquantitativestabilitycurvature-dimensionconditionWassersteindistancerearrangementAubin-Talentibubblesspectralgaplog-Sobolev
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 proves that in a metric measure space with Ricci curvature bounded below (specifically, an essentially non-branching CD(N-1,N) space), any function that nearly saturates the sharp Sobolev inequality must have its distribution of values---the pushforward of the volume measure by the function---close, in Wasserstein distance, to the distribution produced by an Aubin-Talenti bubble on the round sphere. The bound is of order the square root of the deficit epsilon, and this exponent is sharp. The same method yields analogous quantitative stability results for the logarithmic Sobolev inequality and for spectral gap bounds, under various curvature-dimension conditions including infinite and negative effective dimensions. The key mechanism is a rearrangement argument: the function is symmetrized onto a one-dimensional model space, where quantitative stability can be established via spectral analysis or known Euclidean stability results, and the equimeasurability property of rearrangement ensures that the pushforward distribution is preserved under this symmetrization, transferring the one-dimensional stability estimate back to the original space.

What carries the argument

Polya-Szego rearrangement on metric measure spaces; equimeasurability preserving pushforward measures (Lemma 3.5); sharp isoperimetric inequalities in CD spaces; one-dimensional quantitative stability via spectral gap isolation (Theorems 5.2-5.4) and via stereographic projection from known Euclidean results (Proposition 4.2); Wasserstein distance as the stability metric

What would settle it

A space satisfying the curvature-dimension condition and essential non-branching, with a function u that nearly saturates the Sobolev inequality at deficit epsilon but whose pushforward distribution stays farther than C*sqrt(N*epsilon) from every Aubin-Talenti distribution mu_A, for any fixed constant C. The sharpness examples in Section 4.3 show this cannot happen for epsilon^alpha with alpha > 1/2, confirming the exponent is tight.

Watch

Extended reading notes

Core claim

The central object is the pushforward measure u-sharp-m: the probability distribution of values taken by a function u on the space, induced by the volume measure m. The paper shows that when u nearly saturates the sharp Sobolev inequality with deficit epsilon, this pushforward measure is within distance C*sqrt(N*epsilon) (in the 2*-Wasserstein metric) of the family of Aubin-Talenti distributions mu_A on the real line. The reduction works because the Polya-Szego rearrangement of u onto a one-dimensional weighted interval preserves the pushforward measure exactly (Lemma 3.5), while not increasing the Dirichlet energy, so the deficit is preserved and the problem reduces to one-dimensional quant

Load-bearing premise

The argument reduces the problem to a one-dimensional model space using a rearrangement principle that preserves L^p norms and does not increase the Dirichlet energy, but which discards pointwise information about the function. The Wasserstein-distance stability is then deduced from an L^{2*}-norm stability estimate on the rearranged function via the trivial bound that Wasserstein distance is at most the L^2 norm, which may not be tight. The entire approach also requires a Lé

Editorial extensions

If this is right

  • The distributional stability approach provides a template for obtaining quantitative stability of other sharp functional inequalities under curvature bounds, wherever a rearrangement principle and a one-dimensional model are available.
  • The sharp sqrt(epsilon) rate for spectral gap stability under the weaker Rayleigh-quotient assumption (as opposed to requiring u to be an exact eigenfunction) clarifies the distinction between spectral and variational near-extremality.
  • The method extends quantitative stability beyond smooth Riemannian manifolds to the synthetic CD/RCD setting, including singular spaces where geometric stability fails.
  • The approach to the log-Sobolev inequality in the infinite-dimensional limit (RCD(1,infinity)) connects to stability of Gaussian measure and may inform quantitative convergence results in metric measure geometry.

Reading between the lines

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

  • The distributional (Wasserstein) stability is strictly weaker than pointwise or W^{1,2} functional stability, since it only controls the law of u rather than u itself; this suggests that stronger stability results would require additional geometric control on the ambient space beyond what rearrangement provides.
  • The coupling bound W_{2*} <= L^{2*} used to transfer L^{2*}-norm stability to Wasserstein stability is likely not tight; improved transport inequalities could yield better constants or rates, though the exponent 1/2 is already shown to be sharp.
  • The restriction to essentially non-branching spaces is tied to the validity of the isoperimetric inequality; extending to spaces where branching occurs would require a fundamentally different rearrangement principle.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. This paper establishes quantitative stability results for the sharp Sobolev inequality, the logarithmic Sobolev inequality, and the spectral gap under curvature-dimension conditions. The main result (Theorem 1.1) states that on an essentially non-branching CD(N-1,N) space with unit volume, any near-extremizer of the Sobolev inequality has a pushforward distribution close (in Wasserstein distance) to that of an Aubin-Talenti bubble, with deficit bounded by C√(Nε) and sharp exponent 1/2. The strategy is modular: (1) a Pólya-Szegő rearrangement reduces the problem to a one-dimensional model space while preserving pushforward measures (Lemma 3.5); (2) one-dimensional functional stability (Theorem 4.1) is deduced from the Euclidean stability result of [25] via stereographic projection and Hardy-Littlewood rearrangement (Proposition 4.2); (3) the trivial coupling bound W_{2*} ≤ L^{2*} converts functional closeness to distributional closeness. Analogous results are obtained for the LSI (Theorem 1.3) and the spectral gap under CD(N-1,N), RCD(1,∞), and CD(N+1,-N) conditions (Theorem 5.1). Sharpness of the √ε rate is verified by explicit examples in Section 4.3.

Significance. The paper makes a solid contribution by providing the first quantitative stability result for the Sobolev inequality on CD spaces measured in Wasserstein distance of pushforward measures. The approach via rearrangement is elegant and avoids the compactness-based arguments of prior qualitative results. Key strengths include: (i) the parameter-free derivation—the constant β originates from the independent Euclidean stability result [25, Theorem 1.1], and the isoperimetric inequalities are cited from [15, 2, 44] by different author groups; (ii) explicit sharpness examples (Section 4.3, Examples 1 and 2) confirming the optimality of the √ε exponent; (iii) the spectral gap results (Section 5) with sharp linear-in-ε deficit dependence via elementary spectral theory. The extension to non-integer N (Remark 1.2) is noted but without explicit N-dependence, which is appropriately flagged as a limitation.

major comments (2)
  1. Proposition 4.2, proof (p. 9, between equations 4.4 and 4.5): The claim that the infimum in [25, Theorem 1.1] can be restricted to bubbles centered at the origin for radially monotone f relies on the Hardy-Littlewood inequality applied to ∫ f · g_{x0,c,d}^{2*-1}. The argument shows that replacing g_{x0,c,d} by g_{O,c,d} increases this integral, which combined with (4.4) gives (4.5). However, the logic as written is slightly circular: (4.4) computes the distance to a specific g_{x0,c,d}, and the Hardy-Littlewood step shows that the cross-term is maximized at the origin, but one also needs that the self-terms (∥∇g_{x0,c,d}∥² and the L^{2*} norm) are independent of x0. This independence is true for the Euclidean bubbles but should be stated explicitly for the argument to be complete. This is a presentation gap rather than a mathematical error, but it is load-bearing for Proposition 4.2 and,
  2. Theorem 5.1, part iii) (p. 15, equation 5.5): The stability bound W₂(u♯m, ν_{-N}) ≤ √(24ε/((N-1)∧1)³) is stated for CD(N+1,-N) manifolds. The proof (Theorem 5.4) uses spectral analysis of a Schrödinger operator with potential V(t) = (N+1)²/4 - (N+1)(N+3)/4 · cosh²(t), whose essential spectrum starts at (N+1)²/4. The spectral gap δ between the bottom eigenvalue N and the essential spectrum is computed as δ = (N-1)²/4 for 1<N<3 and |N-2| for N≥3. The final constant 16/((N-1)∧1)³ is then obtained from 2N/(δ(N-1)). The case N→1⁺ is singular (δ→0), and the bound diverges. This is consistent with the spectral picture, but the manuscript should briefly comment on whether the rate √ε remains sharp in this regime or whether the dimensional constant degenerates in a way that affects the qualitative conclusion.
minor comments (8)
  1. Remark 1.2: The extension to non-integer N>2 is mentioned but the dependence on N is lost. A brief explanation of where exactly the argument in [25] uses the integer structure (e.g., specific symmetries of R^N) would help the reader understand the obstruction.
  2. Section 4.2, proof of Theorem 1.1 (between equations 4.16 and 4.12): The normalization step from general (a,b) to normalized bubbles (ā, ±√(ā²-1)) is correct via (2.3), but the notation ā is introduced without explicit definition. Adding 'where ā is determined by ∥v_{ā,±√(ā²-1)}∥_{L^{2*}} = 1' would improve readability.
  3. Proposition 4.2, proof: The sign-changing extension is handled by the phrase 'arguing exactly as in [25, Section 3.2]'. While this is a standard reference, a one-sentence summary of the key step (e.g., decomposition into positive and negative parts and separate application of the radially monotone result) would make the proof more self-contained.
  4. Theorem 5.1, part iii): The sharpness of the √ε rate is claimed for all cases but the example for case iii) is only sketched ('Similar examples can be built...'). A brief explicit construction or reference would strengthen the claim.
  5. Section 5, proof of Theorem 5.1: Only case i) is proved, with cases ii) and iii) described as 'completely analogous'. Given that the model space analysis for iii) involves a Schrödinger operator (Theorem 5.4) rather than elementary spectral theory, a brief indication of how the rearrangement step interacts with the unbounded domain I_{-N} = R would be helpful.
  6. Typo in Section 1, line 2 of page 2: 'crucially relyies' should be 'crucially relies'.
  7. The bibliography entry [5] appears to be a duplicate of [4] (both Aubin 1976). These should be merged if appropriate.
  8. Notation: the paper uses both m(X) and m(X)=1 interchangeably. A brief remark that all spaces are assumed to have unit total measure would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment. Both major comments are well-taken presentation issues that we will address in the revision. The first concerns an incomplete justification in the proof of Proposition 4.2 (the translation-invariance of the Euclidean bubble norms), and the second concerns a remark on the degeneracy of the dimensional constant in Theorem 5.1(iii) as N approaches 1. Neither comment identifies a mathematical error; both request clarifications that we are happy to incorporate.

read point-by-point responses

  1. Referee: Proposition 4.2, proof (p. 9, between equations 4.4 and 4.5): The claim that the infimum in [25, Theorem 1.1] can be restricted to bubbles centered at the origin for radially monotone f relies on the Hardy-Littlewood inequality applied to ∫ f · g_{x0,c,d}^{2*-1}. The argument shows that replacing g_{x0,c,d} by g_{O,c,d} increases this integral, which combined with (4.4) gives (4.5). However, the logic as written is slightly circular: (4.4) computes the distance to a specific g_{x0,c,d}, and the Hardy-Littlewood step shows that the cross-term is maximized at the origin, but one also needs that the self-terms (∥∇g_{x0,c,d}∥² and the L^{2*} norm) are independent of x0. This independence is true for the Euclidean bubbles but should be stated explicitly for the argument to be complete.

    Authors: The referee is entirely correct. The translation-invariance of the self-terms — namely that ∥∇g_{x0,c,d}∥²_{L²(R^N)} and ∥g_{x0,c,d}∥_{L^{2*}(R^N)} are independent of the center x₀ — is a necessary ingredient for the Hardy-Littlewood step to yield (4.5), and this fact is currently used but not stated in the manuscript. We will add an explicit sentence between (4.4) and (5.5) noting that, since g_{x0,c,d}(x) = c(d + |x - x₀|²)^{-(N-2)/2} depends on x only through |x - x₀|, both ∥∇g_{x0,c,d}∥²_{L²} and ∥g_{x0,c,d}∥_{L^{2*}}^{2*} are independent of x₀ by translation-invariance of the Lebesgue measure. This makes the logic complete: the only x₀-dependent term in (4.4) is the cross-term ∫ f g_{x0,c,d}^{2*-1}, which is maximized at the origin by Hardy-Littlewood, and therefore the full expression ∥∇f - ∇g_{x0,c,d}∥² is minimized at x₀ = O. We agree this is a presentation gap and will fix it. revision: yes

  2. Referee: Theorem 5.1, part iii) (p. 15, equation 5.5): The stability bound W₂(u♯m, ν_{-N}) ≤ √(24ε/((N-1)∧1)³) is stated for CD(N+1,-N) manifolds. The proof (Theorem 5.4) uses spectral analysis of a Schrödinger operator with potential V(t) = (N+1)²/4 - (N+1)(N+3)/4 · cosh²(t), whose essential spectrum starts at (N+1)²/4. The spectral gap δ between the bottom eigenvalue N and the essential spectrum is computed as δ = (N-1)²/4 for 1<N<3 and |N-2| for N≥3. The final constant 16/((N-1)∧1)³ is then obtained from 2N/(δ(N-1)). The case N→1⁺ is singular (δ→0), and the bound diverges. This is consistent with the spectral picture, but the manuscript should briefly comment on whether the rate √ε remains sharp in this regime or whether the dimensional constant degenerates in a way that affects the qualitative conclusion.

    Authors: We agree that a brief comment on the N → 1⁺ regime is warranted. The situation is as follows: the √ε rate in the deficit remains sharp for each fixed N > 1 (as verified by the explicit examples mentioned after Theorem 5.1, which adapt to the CD(N+1,-N) setting). What degenerates as N → 1⁺ is only the dimensional prefactor 16/((N-1)∧1)³, which diverges because the spectral gap δ between the bottom eigenvalue N and the essential spectrum of the Schrödinger operator L collapses to zero. This is an intrinsic feature of the spectral picture: as N → 1⁺, the eigenvalue N approaches the threshold (N+1)²/4 of the essential spectrum, so the spectral isolation of the ground state degenerates. The qualitative conclusion (convergence of the pushforward to ν_{-N} as ε → 0) remains valid for each fixed N > 1, but the quantitative bound becomes vacuous in the limit N → 1⁺. We will add a remark to this effect after the proof of Theorem 5.4, clarifying that the degeneracy is a spectral-theoretic phenomenon and does not affect the sharpness of the exponent for fixed N. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity found; derivation is modular and parameter-free

full rationale

The paper's central result (Theorem 1.1) is derived through a transparent, modular chain: (1) Pólya-Szegő rearrangement (Theorem 3.1, from [50]) reduces the problem to a 1D model space while preserving pushforward measures (Lemma 3.5, elementary); (2) one-dimensional L^{2*}-stability (Theorem 4.1) is deduced from [25, Theorem 1.1] via stereographic projection and Hardy-Littlewood rearrangement (Proposition 4.2); (3) the trivial coupling bound W_{2*} ≤ L^{2*} (Eq. 4.13) converts functional closeness to distributional closeness. No fitted parameters appear anywhere. The constant β is imported from [25], an independent peer-reviewed result by a different author group (Dolbeault, Esteban, Figalli, Frank, Loss) with an explicit universal value. The isoperimetric inequalities (Theorems 3.2–3.4) are cited from [15, 2, 44] by entirely different author groups. The self-citation [50] (Nobili–Violo) provides the rearrangement framework, but this is a structural tool whose validity does not depend on the target result being proved here; it is independently verifiable and does not assume the conclusion. The sharpness examples (Section 4.3) are constructed explicitly and confirm the √ε rate independently of the upper bound. The normalization step from general (a,b) to normalized bubbles (ā, ±√(ā²−1)) in the proof of Theorem 1.1 is a direct computation via (2.3), not a fit. The only self-citation that could be considered load-bearing is [50] for the rearrangement principle, but since it provides a tool rather than the stability result itself, and since the stability content comes from the independent [25], this is at most a minor self-citation that does not undermine the derivation. No step reduces to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted or introduced ad hoc. The constant C in the main theorem is expressed in terms of the universal constant β from [25], which is explicit and not fitted to the present results. No new mathematical objects, particles, or entities are postulated. All axioms are standard results from the cited literature, verified independently of the present paper.

assumptions (4)
  • domain assumption Sharp isoperimetric inequality holds in CD(N-1,N) spaces (Theorem 3.2, from [15])
    Load-bearing: the Pólya-Szegő rearrangement (Theorem 3.1) requires condition (Iso), which is verified via this isoperimetric inequality. Without it, the energy does not decrease under rearrangement and the entire reduction fails.
  • domain assumption Quantitative stability of Sobolev inequality on R^N with explicit constant β (Theorem 4.2, from [25])
    Load-bearing: the 1D stability (Theorem 4.1) is deduced from this result via stereographic projection. The constant β propagates through to the final estimate.
  • domain assumption Essential non-branching of the metric measure space (Definition 2.2)
    Required for the isoperimetric inequality and rearrangement to hold. Satisfied by RCD spaces and Riemannian manifolds but not all CD spaces.
  • standard math Spectral gap is an isolated eigenvalue of the Laplacian on model spaces (used in Theorems 5.2–5.4)
    Standard spectral theory on 1D weighted intervals. The gap between λ_1 and λ_2 controls the stability constant.

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Pith. "Pith review of Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition." pith.science (2026). https://pith.science/paper/XABY2CWZ

@misc{pith2026260705304,
  author       = {Pith},
  title        = {Pith review of: Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XABY2CWZ}},
  note         = {Machine review of arXiv:2607.05304}
}
abstract

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.

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