An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures

Anqiang Zhu; Kui Wang; Xiaomei Sun

REVIEW 3 major objections 5 minor 22 references

An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For radial log-concave weights on space forms, the ball uniquely minimizes the sum of reciprocals of the first $n$ nonzero Witten-Laplacian Neumann eigenvalues among origin-symmetric Lipschitz domains.

desk verdict New sharp harmonic-mean inequality for Witten-Laplacian Neumann eigenvalues under general convex radial weights; proof mostly sound but Lemma 3.1 has a repairable gap. read the letter →

arxiv 2608.08748 v1 pith:WQ3OUZR5 submitted 2026-08-09 math.SP math.AP

Kui Wang , Xiaomei Sun , Anqiang Zhu This is my paper · ORCID

classification math.SPmath.AP MSC 35P1535J0533A40

keywords NeumanneigenvaluesWitten-Laplacianharmonicmeanisoperimetricinequalitylog-concavemeasurespaceformsSzegő-Weinbergerreciprocaleigenvaluesums

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

The paper proves a sharp isoperimetric inequality for the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian (a Laplacian with a drift induced by the weight) on origin-symmetric Lipschitz domains in the sphere, Euclidean space, and hyperbolic space, equipped with a radial log-concave measure. The central statement is that, under a convexity-plus-derivative condition on the radial weight, the ball of the same weighted volume uniquely minimizes the sum of the reciprocals of the first $n$ nonzero eigenvalues. This simultaneously yields a Szegő-Weinberger type bound: the ball maximizes the first nonzero eigenvalue. The main advance is that the weight no longer has to be non-increasing, so the Gaussian measure and many other radial log-concave weights are covered in a single framework.

What carries the argument

The carrying object is the first nonzero eigenfunction $T_1(r)$ of the Witten-Laplacian on a geodesic ball, together with the sine-type radius function $S_\kappa(r)$ ($\sin r$, $r$, or $\sinh r$) and its derivative $C_\kappa=S_\kappa'$. Lemma 3.3 is the load-bearing step: under the condition $((C_\kappa/S_\kappa)\phi')'\ge 0$, the ratio $T_1(r)/S_\kappa(r)$ is decreasing on $(0,R)$. This monotonicity makes the accumulated function $A$ convex and $H$ strictly concave, so a bathtub-type rearrangement principle turns radial-slice integrals into affine bounds in the volume variable; those bounds become the matrix inequalities $J\succeq aI+cZ$ and $K\preceq \mu_1(B_R)aI-dZ$ with $\mathrm{tr}\,Z=0$. A matrix trace inequality then yields $\mathrm{tr}(K^{-1}J)\ge n/\mu_1(B_R)$, which, combined with the variational principle for reciprocal eigenvalue sums, gives the theorem.

What would settle it

Numerically compute the first eigenfunction on a geodesic ball for a smooth convex radial weight violating $((C_\kappa/S_\kappa)\phi')'\ge 0$, such as $\phi(r)=e^r$ on a small Euclidean ball, and check whether $T_1(r)/S_\kappa(r)$ is decreasing on $(0,R)$. If it increases somewhere, Lemma 3.3 is false and the proof collapses; if it stays monotone, the hypothesis is stronger than needed.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: let $M_\kappa$ be a space form (the sphere for $\kappa=1$, Euclidean space for $\kappa=0$, hyperbolic space for $\kappa=-1$), let $d\gamma_\kappa=e^{-\phi(r)}\,d\mathrm{Vol}_\kappa$ be a radial log-concave measure, and let $\Omega$ be a connected origin-symmetric Lipschitz domain with the same weighted volume as the origin-centered geodesic ball $B_R$. If $\phi$ is convex and $((C_\kappa/S_\kappa)\phi')'\ge 0$ on $(0,R)$, where $S_\kappa(r)$ is $\sin r$, $r$, or $\sinh r$ and $C_\kappa=S_\kappa'$, then $\sum_{i=1}^n 1/\mu_i(\Omega)\ge n/\mu_1(B_R)$, with equality if and only if $\Omega=B_R$. In particular $\mu_1(\Omega)\le \mu_1(B_R)$: among such domains the ball is the unique maximizer of the first nonzero Witten-Laplacian Neumann eigenvalue and the unique minimizer of the sum of reciprocals of the first $n$ nonzero eigenvalues. The proof achieves this without any non-increasing assumption on $\phi$, extending prior results that required monotone weights or the special structure of the Gaussian density.

Load-bearing premise

The load-bearing premise is the monotonicity lemma: under $((C_\kappa/S_\kappa)\phi')'\ge 0$, the ratio $T_1(r)/S_\kappa(r)$ of the first ball eigenfunction to the sine-type radius is decreasing on $(0,R)$, and every later comparison—convexity of $A$, concavity of $H$, and the two matrix bounds—depends on this fact.

Editorial extensions

If this is right

Among origin-symmetric Lipschitz domains of fixed weighted volume satisfying the hypotheses, the ball is the unique maximizer of the first nonzero Witten-Laplacian Neumann eigenvalue.
The Gaussian-space reciprocal-sum inequalities for the first $n$ nonzero eigenvalues follow as a special case, since the Gaussian weight satisfies the derivative condition with equality.
The condition is satisfied by many convex weights, including $\phi(r)=r^{2k}$ with $k\ge 1$ in Euclidean space, so the result is not confined to monotone or Gaussian weights.
The inequality $n/\mu_1(B_R)\le \sum_{i=1}^n 1/\mu_i(\Omega)\le n/\mu_1(\Omega)$ gives two-sided control: the ball eigenvalue bounds the first eigenvalue of every admissible domain.

Reading between the lines

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

The monotonicity condition $((C_\kappa/S_\kappa)\phi')'\ge 0$ may be more than a proof artifact: it is exactly what makes $T_1/S_\kappa$ monotone. Testing smooth convex weights that violate it, such as $\phi(r)=e^r$ on small Euclidean balls, could reveal whether the ball remains extremal under weaker hypotheses.
The same trial-function construction should transfer to any rotationally symmetric model space whose radial function satisfies the identities used here, so the constant-curvature setting is likely not essential.
Using higher spherical harmonics as trial functions might extend the reciprocal-sum bound to more than $n$ eigenvalues, provided the matrix trace inequality admits a generalized form.

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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 paper proves that, for origin-symmetric Lipschitz domains in space forms with radial log-concave measure, the ball minimizes the harmonic mean of the first n nonzero Witten-Laplacian Neumann eigenvalues (equivalently maximizes μ1), under convex φ and ((Cκ/Sκ)φ')' ≥ 0. That genuinely extends Chen–Mao, which needed φ non-increasing, and recovers the Gaussian results of Chiacchio–Di Blasio, Gao–Wang, and Chiacchio. The condition covers φ = r^{2k} on R^n, so this is not a cosmetic generalization. The proof is the right kind: separation on balls, a comparison λ1 < τ2, a monotonicity lemma for T1/Sκ, then a Hersch-type trace inequality with the He–Li–Tang matrix inequality. The structure is coherent and the main theorem looks correct.

Credit where due: the authors identify the correct convexity condition on the weight, and the monotonicity lemma (Lemma 3.3) is the technical heart; the maximum principle argument is plausible and the examples are useful.

Now the soft spots. The most serious is in Lemma 3.1. To conclude μ1(B_R) = λ1, they only show λ1 < τ2, where τ2 is the second k = 0 radial eigenvalue. They never compare λ1 with the first radial eigenvalues for k ≥ 2. The comparison is true — the radial problem's eigenvalue increases with k(k+n−2) — but it is not written. As is, the proof of the multiplicity and form of the first eigenfunctions is incomplete. This is repairable, but it is load-bearing because the rest of the paper uses T1 and its monotonicity.

Second, the boundary argument at r = 0 uses Sκ(0) = 0 and drops terms; that requires n ≥ 2. The theorem never states n ≥ 2, though the n = 1 case is trivial. Minor but should be stated.

Third, identity (3.15) is asserted after "further simplification" — the reader cannot verify it without doing the algebra. Given the importance of that identity, showing the steps would help. There are also two notational slips: in the displayed definition of d they write G(R)^2/R^2 instead of G(R)^2/Sκ(R)^2 (only the same for κ = 0), and (4.9) writes Sκ(r)^2 where the context needs Sκ(r)^{n-1}. These are typos, not mathematical errors.

The reliance on Lemma 2.1 from an unpublished preprint [12] is a citation concern, but the inequality itself is plausible and, if needed, can be proved separately; it is not a defect of this paper's argument.

Overall: the main theorem is very likely correct, and the proof strategy is sound modulo the Lemma 3.1 gap. An editor should send this to a serious referee. I would bring it to the reading group and would cite it once the gap is closed.

Referee Report

3 major / 5 minor

Summary. The manuscript proves a sharp isoperimetric inequality for the harmonic mean of the first n nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, under a radial log-concave measure dγ_κ = e^{-φ(r)} dVol_κ. The main theorem, Theorem 1.1, states that if φ is convex and ((C_κ/S_κ)φ')' ≥ 0 on (0,R), then for Ω ∈ E_κ with the same weighted volume as the centered ball B_R, one has ∑_{i=1}^n 1/μ_i(Ω) ≥ n/μ_1(B_R), with equality only for the ball. The proof combines separation of variables on balls, monotonicity properties of the first radial eigenfunction T_1, the bathtub principle for radial rearrangements, a Hersch-type trace formulation, and a matrix trace inequality from He-Li-Tang. The result is presented as an extension of Gaussian and Euclidean results by Chiacchio, Gao-Wang, and Chen-Mao.

Significance. If correct, Theorem 1.1 gives a substantial unified sharp inequality for a broad class of weighted manifolds, removing the non-increasing assumption on φ that was present in prior work. It recovers known Gaussian-space inequalities and yields a new family of Szegő-Weinberger-type bounds. The proof strategy is conceptually clear and the paper is careful in connecting the monotonicity of T_1/S_κ to matrix inequalities. The main strengths are the parameter-free sharp inequality, the explicit equality statement, and the breadth of admissible weights (for example, φ(r)=r^{2k} in R^n). However, the current version has a load-bearing gap in the identification of the first eigenfunction on balls and relies on an unpublished lemma for the decisive trace step, so the verification is not yet complete.

major comments (3)

[Section 3, Lemma 3.1] The proof establishes λ1 < τ2, where τ2 is the second radial eigenvalue for k=0, and then immediately concludes that μ1(B_R)=λ1. This conclusion is not justified, because the separated spectrum also contains, for each angular quantum number k≥2, the first radial eigenvalues λ_{k,1} of the problems T'' + ((n-1)C_κ/S_κ - φ')T' + (μ - k(k+n-2)/S_κ^2)T = 0. The paper never compares λ1 with λ_{k,1} for k≥2. This comparison is load-bearing: the subsequent Lemmas 3.2 and 3.3 and the trial functions in Section 4 all use that T_1 is a genuine first eigenfunction. The gap is repairable: the Rayleigh quotient for the k-th angular mode, namely [∫ p(T')^2 + k(k+n-2)∫ p T^2/S_κ^2]/∫ p T^2, is increasing in k, so λ1 ≤ λ_{k,1} for all k≥1; this argument should be included explicitly.
[Section 2, Lemma 2.1] The decisive matrix inequality (2.2), used to pass from the bounds J ⪰ aI + cZ and K ⪯ λaI - dZ to the final trace lower bound, is quoted from the unpublished preprint [12] by He-Li-Tang. Because this lemma is the central engine of the proof and its equality statement is also used, the manuscript should provide a proof in an appendix or cite a peer-reviewed version. As it stands, a reader cannot verify the key inequality except by trusting an unpublished source.
[Throughout, Theorem 1.1 and Lemmas 3.1–3.3] The paper does not state n≥2, but several arguments rely on n≥2. In particular, Lemma 3.2 drops the boundary term at r=0 using S_κ(0)=0, which fails for n=1 because S_κ(0)^{n-1}=1; Lemma 3.1's multiplicity-n statement is also false in dimension one. The n=1 case is presumably trivial or otherwise known, but it should be explicitly excluded or treated separately.

minor comments (5)

[Section 3, Lemma 3.3, identity (3.15)] The derivation of (3.15) from (3.14) is omitted. The identity is correct after using (C_κ/S_κ)' = -1/S_κ^2 and collecting terms, but this algebra is central to the maximum-principle argument and should be displayed or at least sketched.
[Section 4, inequality (4.9)] In (4.9) the factor S_κ(r)^2 should presumably be S_κ(r)^{n-1} to match the polar-coordinate integration dγ_κ = S_κ^{n-1}e^{-φ} dr dθ and the subsequent expression in (4.12); as printed the inequality is dimensionally inconsistent.
[Section 4, after (4.5)–(4.6)] The text says that H'(y) is strictly decreasing, but Lemma 3.3 only establishes that T_1/S_κ is monotonically decreasing, i.e. non-increasing. If strictness is needed for the equality case or for the strict version of the bathtub principle, it should be proved or the statements should be weakened to non-strict monotonicity.
[Section 3, Lemma 3.2] The sentence 'Hence, in all cases under consideration, for every r∈(0,R), 0>∫_0^r (μ-(n-1)/S_κ^2)T_1 p dr' is not self-evident; it relies on the fact that the integrand's indefinite integral has total mass zero at r=R and that μ-(n-1)/S_κ^2 is strictly increasing, forcing the cumulative integral to be strictly negative for r∈(0,R). This one-sentence justification would improve readability.
[Throughout] There are several typos, including 'muliplicity' and 'eigfenfunction' in Lemma 3.1 and 'domian' in Section 2; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and does not reduce to its inputs.

full rationale

The paper proves a sharp isoperimetric inequality for reciprocal sums of Witten-Laplacian Neumann eigenvalues. The proof is genuinely self-contained in the relevant analytic sense: it constructs trial functions from the first ball eigenfunction T1(r)ψi(θ), establishes the needed monotonicity of T1/Sκ in Lemma 3.3 under the stated condition ((Cκ/Sκ)φ′)′ ≥ 0, and then applies the external matrix trace inequality of He–Li–Tang (Lemma 2.1), Hersch's variational principle (Lemma 2.2), and the bathtub principle (Lemma 2.3). These are cited as external tools, not derived from, or equivalent to, the target theorem. No parameter is fitted to data and no predicted quantity is defined in terms of the claimed inequality. The self-citations [11], [17], and [19] are contextual references to prior work and are not invoked as load-bearing steps in the proof of Theorem 1.1. Even though the skeptic's concern about Lemma 3.1 (the missing comparison with angular modes k≥2) may indicate a repairable gap in the proof of the ball eigenvalue structure, that is a correctness or completeness issue, not circularity, because the paper does not assume the conclusion of the theorem to establish it. Consequently, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: R is determined by volume equality |Ω| = |B_R|. The proof introduces no new objects beyond the standard trial functions built from ball eigenfunctions. The listed axioms are standard tools cited from prior work; the main new ingredient is Lemma 3.3, whose proof is internal.

assumptions (5)

standard math Lemma 2.1 matrix trace inequality (He-Li-Tang, arXiv 2606.08271) is valid.
Used in Section 4 after (4.13) to conclude tr(K^{-1}J) ≥ n/μ1(B_R). The paper states it without proof.
standard math Hersch variational principle for reciprocal sums (Lemma 2.2).
Gives the starting inequality sum 1/μ_i ≥ tr(K^{-1}J) for mean-zero trial functions; cited to [13, 9, 14].
standard math Bathtub principle for radial rearrangement (Lemma 2.3).
Used in Section 4 to compare integrals over radial slices E_θ with integrals over initial intervals of the same weighted measure.
domain assumption Compact embedding H^1(Ω,γκ) into L^2(Ω,γκ) (equation 2.1).
Assumed for the spectrum to be discrete; cited to Chiacchio-Gavitone [10].
standard math Standard Sturm-Liouville comparison and separation of variables on geodesic balls.
Basis for Lemmas 3.1 and 3.2, including multiplicity of the first nonzero eigenvalue on balls.

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Cite this review

Pith. "Pith review of An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures." pith.science (2026). https://pith.science/paper/WQ3OUZR5

@misc{pith2026260808748,
  author       = {Pith},
  title        = {Pith review of: An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQ3OUZR5}},
  note         = {Machine review of arXiv:2608.08748}
}

abstract

We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [12]

A proof of the Ashbaugh-Benguria conjecture for reciprocal sums of Neumann eigenvalues.arXiv: 2606.08271, 2026

Yixin He, Yanyang Li, and Quanyu Tang. A proof of the Ashbaugh-Benguria conjecture for reciprocal sums of Neumann eigenvalues.arXiv: 2606.08271, 2026

arXiv 2026

[2] [1]

A. R. Aithal and G. Santhanam. Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces.Trans. Amer. Math. Soc., 348(10):3955–3965, 1996

work page 1996

[3] [2]

Ashbaugh

Mark S. Ashbaugh. Open problems on eigenvalues of the Laplacian. InAnalytic and geometric in- equalities and applications, volume 478 ofMath. Appl., pages 13–28. Kluwer Acad. Publ., Dordrecht, 1999

work page 1999

[4] [3]

Ashbaugh and Rafael D

Mark S. Ashbaugh and Rafael D. Benguria. Universal bounds for the low eigenvalues of Neumann Laplacians inndimensions.SIAM J. Math. Anal., 24(3):557–570, 1993

work page 1993

[5] [4]

Ashbaugh and Rafael D

Mark S. Ashbaugh and Rafael D. Benguria. Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature.J. London Math. Soc. (2), 52(2):402–416, 1995

work page 1995

[6] [5]

Benguria, Barbara Brandolini, and Francesco Chiacchio

Rafael D. Benguria, Barbara Brandolini, and Francesco Chiacchio. A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere.Commun. Contemp. Math., 22(3):1950018, 9, 2020

work page 2020

[7] [6]

Optimal Szeg¨ o-Weinberger type inequalities.Commun

Friedemann Brock, Francesco Chiacchio, and Giuseppina di Blasio. Optimal Szeg¨ o-Weinberger type inequalities.Commun. Pure Appl. Anal., 15(2):367–383, 2016

work page 2016

[8] [7]

On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian.arXiv: 2403.08070, 2024

Ruifeng Chen and Jing Mao. On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian.arXiv: 2403.08070, 2024

arXiv 2024