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A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

T0 review · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The ball uniquely minimizes the sum of the reciprocals of the first m nonzero Neumann eigenvalues among smooth bounded domains of fixed volume.

desk verdict The paper claims a complete proof of the Ashbaugh-Benguria conjecture for the sum of reciprocals of the first m Neumann eigenvalues on smooth domains. read the letter →

arxiv 2606.08271 v1 pith:CILJQBR4 submitted 2026-06-06 math.SP math.AP

Yixin He , Yanyang Li , Quanyu Tang This is my paper · ORCID
classification math.SPmath.AP
keywords NeumanneigenvaluesAshbaugh-Benguriaconjecturedomainoptimizationspectralgeometryeigenvaluesumsballminimizersisoperimetricinequality
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 among all smooth bounded domains in Euclidean space R^m with a fixed volume, the ball achieves the smallest possible value of the sum of the reciprocals of the first m positive Neumann eigenvalues. Equality holds only when the domain is a ball. The result confirms the Ashbaugh-Benguria conjecture under the stated smoothness assumption. A reader cares because the claim identifies the shape that extremizes a concrete combination of vibration frequencies determined by the Neumann boundary condition.

What carries the argument

The functional given by the sum of the reciprocals of the first m positive Neumann eigenvalues, minimized over the class of smooth bounded domains of fixed volume.

What would settle it

Numerically compute the first m nonzero Neumann eigenvalues on a non-ball domain of the same volume as the unit ball (for example an ellipsoid in R^2) and check whether their reciprocal sum is strictly larger than the corresponding sum on the ball.

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Extended reading notes

Core claim

Among all smooth bounded domains of fixed volume in R^m, the ball minimizes the sum of the reciprocals of the first m nonzero Neumann eigenvalues, and equality is attained precisely by balls.

Load-bearing premise

The domains are required to have smooth boundaries.

Editorial extensions

If this is right

  • Equality holds if and only if the domain is a ball.
  • The stated minimization property holds in every dimension m.
  • The result applies to every smooth bounded domain of the given volume.

Reading between the lines

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

  • The same minimization may hold for other spectral functionals or different boundary conditions if the proof technique extends.
  • Approximation arguments might allow passage from smooth to Lipschitz domains without changing the minimizer.
  • Related isoperimetric problems for sums involving higher Neumann eigenvalues could be approachable by the same methods.
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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

0 major / 0 minor

Summary. The manuscript asserts a proof of the Ashbaugh--Benguria conjecture restricted to smooth bounded domains of fixed volume in R^m: the Euclidean ball minimizes the sum of the reciprocals of the first m nonzero Neumann eigenvalues, with equality attained if and only if the domain is a ball.

Significance. If the argument is correct, the result settles a specific case of a well-known conjecture in spectral geometry by establishing an isoperimetric inequality for a reciprocal-sum functional of Neumann eigenvalues. The explicit smoothness hypothesis on the boundary is already incorporated into the statement, avoiding any mismatch with the claimed scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary accurately reflects the scope and results of the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct proof of conjecture

full rationale

The paper states a direct mathematical proof of the Ashbaugh-Benguria conjecture for the reciprocal sum of Neumann eigenvalues on smooth bounded domains. No equations, parameters, or steps are described that reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The result is presented as an independent theorem with equality cases for balls, consistent with a self-contained derivation against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the standard existence and variational properties of the Neumann Laplacian on smooth domains; no free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • standard math Existence, positivity, and variational characterization of the Neumann eigenvalues on smooth bounded domains in R^m
    The statement presupposes the classical spectral theory of the Neumann Laplacian.

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Cite this review

Pith. "Pith review of A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues." pith.science (2026). https://pith.science/paper/CILJQBR4

@misc{pith2026260608271,
  author       = {Pith},
  title        = {Pith review of: A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CILJQBR4}},
  note         = {Machine review of arXiv:2606.08271}
}
abstract

We prove the Ashbaugh--Benguria conjecture for bounded domains with smooth boundary in $\mathbb R^m$. More precisely, among all smooth bounded domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

    math.AP 2026-07 unverdicted novelty 8.0 of 10

    Proves that for bounded domains in R^N (N≥2), the sum from i=1 to N of λ1/(λ_{i+1}-λ1) is at least N/(j_{N/2,1}^2/j_{N/2-1,1}^2 -1), with equality precisely when the domain is a ball up to H^1-capacity zero.

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

    math.SP 2026-08 conditional novelty 7.0 of 10

    For Witten-Laplacians with radial log-concave measures on space forms, the geodesic ball uniquely minimizes the harmonic mean of the first n nonzero Neumann eigenvalues.

  3. Reciprocal sums of Neumann eigenvalues in non-Euclidean space forms

    math.DG 2026-06 unverdicted novelty 7.0 of 10

    Proves sum_{j=1 to n} 1/μ_j(Ω) ≥ n/μ_1(B_Ω) for Neumann eigenvalues on domains in space forms of curvature ±1, equality iff Ω is a geodesic ball.

  4. A sharp Gaussian harmonic-mean inequality for Neumann eigenvalues of the Ornstein-Uhlenbeck operator

    math.SP 2026-07 accept novelty 6.0 of 10

    For origin-symmetric Gaussian domains, the harmonic mean of the first N Neumann Ornstein–Uhlenbeck eigenvalues is maximized uniquely by the centered ball of equal Gaussian measure.

  5. Spectral and Geometric Stability for the Reciprocal Sum of Neumann Eigenvalues

    math.SP 2026-07 conditional novelty 6.0 of 10

    The reciprocal-sum deficit for the first d nonzero Neumann eigenvalues controls eigenvalue displacements, spectral splitting, and Fraenkel asymmetry with sharp, optimal exponents.

Reference graph

Works this paper leans on

18 extracted references · 2 canonical work pages · cited by 5 Pith papers

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Reviewed June 27, 2026 · model on record in the stance chip above.