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arxiv: 2606.08271 · v1 · pith:CILJQBR4new · submitted 2026-06-06 · 🧮 math.SP · math.AP

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

Yixin He , Yanyang Li , Quanyu Tang This is my paper

Pith reviewed 2026-06-27 18:44 UTC · model grok-4.3

classification 🧮 math.SP math.AP
keywords Neumann eigenvaluesAshbaugh-Benguria conjecturedomain optimizationspectral geometryeigenvalue sumsball minimizersisoperimetric inequality
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The pith

The ball uniquely minimizes the sum of the reciprocals of the first m nonzero Neumann eigenvalues among smooth bounded domains of fixed volume.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

The domains are required to have smooth boundaries.

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.

read the original 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.

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 / 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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.1-grok · 5569 in / 1091 out tokens · 21002 ms · 2026-06-27T18:44:40.932868+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

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

    math.AP 2026-07 unverdicted novelty 8.0

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

    math.DG 2026-06 unverdicted novelty 7.0

    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.

Reference graph

Works this paper leans on

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

  1. [1]

    M. S. Ashbaugh,Open problems on eigenvalues of the Laplacian, in: Analytic and Geometric Inequalities and Applications, Math. Appl., vol. 478, Kluwer Academic Publishers, Dordrecht, 1999, pp. 13–28

    1999

  2. [2]

    M. S. Ashbaugh and R. D. Benguria,Universal bounds for the low eigenvalues of Neumann Laplacians in Ndimensions, SIAM J. Math. Anal.24(1993), no. 3, 557–570

    1993

  3. [3]

    Bhatia,Matrix Analysis, Graduate Texts in Mathematics, vol

    R. Bhatia,Matrix Analysis, Graduate Texts in Mathematics, vol. 169, Springer, New York, 1997

    1997

  4. [4]

    Brock,An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z

    F. Brock,An isoperimetric inequality for eigenvalues of the Stekloff problem, ZAMM Z. Angew. Math. Mech.81(2001), no. 1, 69–71

    2001

  5. [5]

    Chen and J

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

    work page arXiv 2024

  6. [6]

    Dittmar,Sums of reciprocal eigenvalues of the Laplacian, Math

    B. Dittmar,Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr.237(2002), 45–61

    2002

  7. [7]

    M.Eddaoudi,Lower bounds for the sum of the reciprocals of eigenvalues of bounded domains inRn, spheres, and closed orientable surfaces, arXiv:2502.13089, 2025

    work page arXiv 2025

  8. [8]

    Enache and G

    C. Enache and G. A. Philippin,Some inequalities involving eigenvalues of the Neumann Laplacian, Math. Methods Appl. Sci.36(2013), no. 16, 2145–2153

    2013

  9. [9]

    Gao and K

    Y. Gao and K. Wang,An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space, J. Funct. Anal.290(2026), no. 8, Paper No. 111379

    2026

  10. [10]

    Henrot,Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006

    A. Henrot,Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006

    2006

  11. [11]

    Hersch,Caractérisation variationnelle d’une somme de valeurs propres consécutives; généralisation d’inégalités de Pólya–Schiffer et de Weyl, C

    J. Hersch,Caractérisation variationnelle d’une somme de valeurs propres consécutives; généralisation d’inégalités de Pólya–Schiffer et de Weyl, C. R. Acad. Sci. Paris252(1961), 1714–1716

    1961

  12. [12]

    G. N. Hile and Z. Y. Xu,Inequalities for sums of reciprocals of eigenvalues, J. Math. Anal. Appl.180 (1993), no. 2, 412–430

    1993

  13. [13]

    R. S. Laugesen,Well-posedness of Weinberger’s center of mass by Euclidean energy minimization, J. Geom. Anal.31(2021), no. 9, 8762–8779

    2021

  14. [14]

    Levitin, D

    M. Levitin, D. Mangoubi, and I. Polterovich,Topics in Spectral Geometry, Graduate Studies in Mathe- matics, vol. 237, American Mathematical Society, Providence, RI, 2023

    2023

  15. [15]

    E. H. Lieb and M. Loss,Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathe- matical Society, Providence, RI, 2001

    2001

  16. [16]

    Szegő,Inequalities for certain eigenvalues of a membrane of given area, J

    G. Szegő,Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal.3 (1954), 343–356

    1954

  17. [17]

    H. F. Weinberger,An isoperimetric inequality for theN-dimensional free membrane problem, J. Rational Mech. Anal.5(1956), 633–636

    1956

  18. [18]

    Xia and Q

    C. Xia and Q. Wang,On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann Laplacian, Math. Ann.385(2023), no. 1–2, 863–879. School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China Email address:yixin.he717@gmail.com School of Mathematics, Southeast University, Nanjing 211189, P. R. China Email address:liyan...

    2023