On Berger's Isoperimetric Problem

Fan Kang

REVIEW 3 major objections 4 minor 15 references

On Berger's Isoperimetric Problem

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper gives a new proof that the flat equilateral torus maximizes the product of the first Laplace eigenvalue and area, settling the isoperimetric problem for tori.

desk verdict New proof of a known theorem and a genuinely improved conformal eigenvalue bound, but with an unshown computer-algebra step and a typo in the displayed corollary that both need fixing before the paper is dependable. read the letter →

arxiv 2506.05775 v1 pith:A5L7X36V submitted 2025-06-06 math.DG math.SP

Fan Kang This is my paper

classification math.DGmath.SP MSC 58J5053C4235P15

keywords Berger'sisoperimetricproblemfirstLaplaceeigenvalueflattorusequilateralconformalareaλ1-maximalsurface

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 gives a new proof that the flat equilateral torus is the unique maximizer of the product of the first Laplace eigenvalue and area among all tori, a result known as Berger's isoperimetric problem. The argument uses a prior energy-ratio method together with a cited computation of conformal area for explicit immersions of tori into the 3-sphere. A corollary improves the known upper bound for the first conformal eigenvalue of an arbitrary flat torus, with strict improvement outside the arc $a^2+b^2=1$ in the moduli region.

What carries the argument

The central object is the conformal-area functional $A(\gamma\circ\psi)$ for a sphere-valued immersion $\psi$, maximized over conformal transformations $\gamma$ of the sphere. A cited prior result (Lemma 2.1) gives the exact value of this supremum for the special immersions $\psi_b$ into $S^3$. The proof then uses an energy-ratio identity that equates energy ratios of sphere-valued maps to area ratios of their conformally transformed versions, together with a centering lemma that makes the coordinate integrals of the test map vanish, so the min-max principle applies.

What would settle it

Rerun the polynomial factorization used in Lemma 2.1 and verify the claimed product form of the basis element; independently, numerically maximize the function $I(\lambda,\mu)$ over the region $\Omega$ for some $b>\sqrt{2}$ (for example $b=2$) and compare with the claimed supremum $8\pi^2\sqrt{b^2+1}/(3\sqrt{3}b)$.

Watch

Extended reading notes

Core claim

The paper proves Theorem 1.1: $\Lambda_1(T)=8\pi^2/\sqrt{3}$, and the flat equilateral torus is the only $\lambda_1$-maximal surface. The proof splits the moduli space of flat tori into two regions and handles each with a test map whose conformal area is known from a cited prior result. The energy-ratio identity (Proposition 2.2) converts this area information into the uniform upper bound $\Lambda_1(T)\leq 8\pi^2/\sqrt{3}$, and equality forces the parameters $(a,b)=(1/2,\sqrt{3}/2)$, the equilateral torus.

Load-bearing premise

The claim that the area functional has no interior critical point rests on an unshown polynomial factorization produced by a computer algebra calculation; if that factorization is wrong, the upper bound proof would break.

Editorial extensions

If this is right

Recovers the known result that the equilateral torus uniquely maximizes $\lambda_1 A$ among all tori.
Improves the existing upper bound for the first conformal eigenvalue of any flat torus, with strict inequality whenever $a^2+b^2>1$ in the fundamental domain.
The equality case is attained exactly at $(a,b)=(1/2,\sqrt{3}/2)$.
The same energy-ratio plus conformal-area strategy could be reused for other compact surfaces once the relevant conformal-area maximization is known.

Reading between the lines

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

If the computer algebra factorization in Lemma 2.1 were published in full, the proof would be checkable without rerunning the computation; currently a skeptical reader must redo the calculation to confirm it.
The strict bound in Corollary 1.2 quantifies a spectral gap between non-equilateral flat tori and the maximum, which could be useful for numerical work on eigenvalue optimization.
One could try the same scheme on the Klein bottle or genus-two surfaces, where analogous maximal metrics are conjectured, provided conformal-area formulas for suitable test maps exist.

Share X Bluesky LinkedIn Reddit HN

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 is a new proof of Nadirashvili's 1996 theorem that the flat equilateral torus maximizes lambda_1 A among all tori, plus an improved upper bound for the first conformal eigenvalue of every flat torus (Corollary 1.2). The main theorem is not new, but the proof is: it uses the El Soufi-Ilias-Ros energy invariance trick and a conformal-area computation from Bryant, rather than the calculus-of-variations existence argument. The corollary's bound is genuinely new and is tighter than the El Soufi-Ilias-Ros bound everywhere in the moduli space, and for the equilateral torus it hits 8 pi^2 / sqrt(3). That is a real improvement.

The proof is structurally sound. The M1 branch (b <= sqrt(2)) is fully analytic; the min-max argument with Hersch's lemma is standard; the energy ratios check out; and for the equilateral torus the bound is attained only by the flat metric. The M2 branch reduces to Lemma 2.1, where the b > sqrt(2) case hinges on a MAPLE Groebner basis computation: the paper writes that the first element factors as G(lambda, mu) and that G is positive on Omega, but it never lists the polynomials U and V, nor shows the computation. That is a load-bearing step. If the factorization or positivity claim is wrong, the no-interior-critical-point conclusion fails and the bound (2.4) is not proven. For a new proof of a known theorem this is not fatal, but a referee should ask for the polynomials and a verification script, or an independent argument.

There is also a typo in the displayed formula (1.3) in Corollary 1.2. As printed, the expression does not reduce correctly for the equilateral torus; the proof later derives the right formula via the minimization of F(b0), and that derivation checks out algebraically. So the issue looks like a typesetting error in the introduction, not a mathematical gap, but it must be fixed.

Minor comment: the strictness argument for a^2+b^2 > 1 is given by a displayed nonzero expression. It is plausible but dense; a referee should verify the sign and the Laplacian computation.

Bottom line: this is an honest, well-cited paper that adds a new pathway to a classical result and an genuinely improved bound. The unshown MAPLE step and the formula typo are fixable, but they must be addressed. I'd send this to a serious referee; the referee should have access to MAPLE or a computer algebra system to check the Groebner claim. Once those pieces are supplied, I'd be happy to cite the corollary.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new proof of Nadirashvili's theorem that the first positive conformal eigenvalue of a torus is maximized by the flat equilateral torus. The proof combines the El Soufi-Ilias-Ros energy-invariance argument with Bryant's computation of the conformal area of the family of immersions ψ_b of flat tori into S^3. For tori with b≤√2 and b>√2, the author constructs explicit test maps into S^3, balances them using Hersch's lemma, and bounds the resulting Dirichlet energy by the supremum of the conformal-area functional. A corollary gives an upper bound for the first conformal eigenvalue in every flat conformal class. The main theorem is reduced to an explicit one-parameter analytic estimate, Lemma 2.1, whose proof is the core technical step.

Significance. If the proof is completed, the paper offers a genuinely different route to a classical rigidity result, avoiding the variational existence argument of Nadirashvili and instead deriving the sharp bound from explicit trigonometric integrals and Bryant's conformal-area formulas. The strategy is attractive and the surrounding reductions are coherent: Hersch balancing, the energy-ratio invariance of Proposition 2.2, and the M1/M2 summation in the proof of Theorem 1.1 are arithmetically consistent, and the argument contains no fitted parameters. The main load-bearing gaps are the unshown Gröbner-basis step in Lemma 2.1 and a concrete inconsistency between the displayed Corollary bound (1.3) and the function F(b0') obtained in the proof.

major comments (3)

[Section 2, Lemma 2.1, Case 2] The claim that I(λ,μ) has no critical point in the interior of Rδ is the essential step that converts the boundary computation into the supremum (2.4). The proof of this claim is not self-contained: the polynomials U(r1,λ,μ) and V(r1,λ,μ) are never written, the MAPLE Gröbner-basis computation is not shown, and the displayed factorization of the first basis element G(λ,μ) is offered without a certificate or reproducible code. Since (2.4) is used directly in the M2 branch of Theorem 1.1, an incorrect or incomplete verification of this factorization would invalidate the main theorem. Please provide U and V together with the actual Gröbner-basis computation (or a reproducible script and output), or state Lemma 2.1 as a black box from Bryant and do not claim a self-contained proof.
[Corollary 1.2, Eq. (1.3)] The displayed formula (1.3) does not agree with the function F(b0') derived in the proof of the corollary. For the equilateral torus (a,b)=(1/2,√3/2), we have a²+b²=1, L=3, and b=√3/2. If the radical covers only the numerator, the right-hand side of (1.3) evaluates to 16π²/(3√3), which is strictly smaller than Λ1(T)=8π²/√3, so the inequality would be false. If the radical covers the whole fraction, the value is 32π²/(3√3), which is not equal to the value F(√2)=8π²/√3 obtained from (2.11). Thus the statement that F(b0') is the right-hand side of (1.3) is incorrect in either reading. The corollary and its strictness proof need to be corrected so that the displayed bound matches the function actually minimized in the proof.
[Proof of Corollary 1.2, strictness paragraph] The strictness argument for a²+b²>1 relies on a displayed Wronskian containing the undefined symbol λ, and the final ''≠0'' is stated without proof. Equality in the chain would imply that the components of γ∘Φ_{b0'} are first eigenfunctions, and the nonzero Wronskian is used to contradict that conclusion. Because the bracket displayed is a sum of trigonometric terms and could vanish at isolated points, the proof should either show explicitly that the expression is not identically zero as a function on the torus, or replace this step with a clearer argument. This is needed for the strictness claim of the corollary.

minor comments (4)

[Introduction and References] The text cites Li-Yau [Yau82] for boundedness of λ1(g)A(g) in a conformal class, but the bibliography's Li-Yau paper is [LY82]; the reference [Yau82] appears to be a different item. Please correct the citation.
[Section 2, Lemma 2.1] The notation Ω is used both for the open ellipse {λ²/r1+μ²/r2<1, λ,μ≥0} and for its closure. Please distinguish, for instance, Ω and the closure of Ω, to avoid ambiguity in the boundary analysis.
[Section 2, Eq. (2.7)] The second definite integral formula is quoted from the appendix of Bryant [Bry15] without proof. If the proof of Lemma 2.1 is meant to be complete, include a derivation or a precise restatement of the formula; otherwise, make explicit that this part of Lemma 2.1 is imported from Bryant.
[Proof of Corollary 1.2] The strictness display contains an undefined λ in the factor 1/(b²λ²), and the parentheses in the trigonometric bracket are difficult to parse. Please rewrite this computation with clear notation and state the not-identically-zero property explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof of Theorem 1.1 rests on independent external results (Bryant's conformal-area computation, El Soufi–Ilias–Ros energy invariance, Hersch's balancing lemma), and the target eigenvalue bound is not assumed as an input.

full rationale

The paper's derivation chain is self-contained with respect to its stated external benchmarks. Theorem 1.1 is proved by combining Hersch's balancing lemma, the El Soufi–Ilias–Ros energy-invariance proposition (Proposition 2.2), and the conformal-area supremum of the immersed torus ψ_b. The latter is quoted from Bryant and then reproved in Lemma 2.1 using explicit integral formulas and a Gröbner-basis computation. The desired final constant Λ_1(T)=8π²/√3 is not inserted anywhere as an input; it emerges after separately bounding the conformal eigenvalue on M1 and M2 and maximizing the resulting elementary functions. The only potentially load-bearing analytic step, the claim that the polynomial G(λ,μ) is positive on the interior of Ω and that the Gröbner basis factors as stated, is an unshown MAPLE computation. That is a verifiability and correctness concern about the proof's rigor, not circularity: the step is not assumed from the theorem being proved and does not reduce the prediction to its own definition. No fitted parameters are renamed as predictions, and no load-bearing result is justified solely by a self-citation from the same author. The acknowledgements cite private correspondence with Robert Bryant and a supervisor, but those are not used as substitutes for mathematical proof in a circular way. Overall, the paper uses external benchmarks rather than assuming its conclusion, so the circularity score is 0.

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

The central claim rests on Bryant's conformal area theorem and the ESIR energy invariance result, both taken from prior literature, plus standard spectral geometry facts. There are no fitted parameters or newly invented entities.

assumptions (4)

domain assumption Bryant's conformal area formula for the immersions psi_b (Lemma 2.1), including the integral formula for the elliptic integral.
The lemma is restated from [Bry15]. The proof in the paper cites an integral formula to Bryant's appendix and uses a MAPLE computation to establish the absence of interior critical points, so the full statement is not derived within the paper.
domain assumption El Soufi-Ilias-Ros energy ratio invariance (Proposition 2.2, from [ESIR96]).
Used without proof to relate the energy of conformally transformed maps to the area of the model immersions. This is a known result in the field.
standard math Hersch's balancing lemma: for any map into a sphere there exists a conformal transformation making the components have zero mean.
Invoked in the proof of Theorem 1.1 to ensure the test functions are admissible for the min-max principle.
standard math Min-max principle for the first eigenvalue of the Laplacian.
Used to convert bounds on test-function energies into bounds on lambda_1(g)A(g).

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Berger's Isoperimetric Problem." pith.science (2026). https://pith.science/paper/A5L7X36V

@misc{pith2026250605775,
  author       = {Pith},
  title        = {Pith review of: On Berger's Isoperimetric Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5L7X36V}},
  note         = {Machine review of arXiv:2506.05775}
}

abstract

Berger's isoperimetric problem asks if the flat equilateral torus is $\lambda_1$-maximal. In 1996, Nadirashvili first gave a positive answer. In this paper, we use El Soufi-Ilias-Ros's method and Bryant's result (arXiv:1507.01485) to give a new proof.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages

[1]

2, 129--149

Marcel Berger, Sur les premieres valeurs propres des vari \'e t \'e s riemanniennes , Compositio Mathematica 26 (1973), no. 2, 129--149

work page 1973
[2]

Robert L Bryant, On the conformal volume of 2-tori, arXiv preprint arXiv:1507.01485 (2015)

arXiv 2015
[3]

4, 667--690

Donato Cianci, Mikhail Karpukhin, and Vladimir Medvedev, On branched minimal immersions of surfaces by first eigenfunctions, Annals of Global Analysis and Geometry 56 (2019), no. 4, 667--690

work page 2019
[4]

Ahmad El Soufi, Hector Giacomini, and Mustapha Jazar, A unique extremal metric for the least eigenvalue of the laplacian on the klein bottle, Duke Mathematical Journal (2006)

work page 2006
[5]

Ahmad El Soufi and Said Ilias, Immersions minimales, premiere valeur propre du laplacien et volume conforme, Mathematische Annalen 275 (1986), 257--267

work page 1986
[6]

5, 1611--1618

, Extremal metrics for the first eigenvalue of the laplacian in a conformal class, Proceedings of the American Mathematical Society 131 (2003), no. 5, 1611--1618

work page 2003
[7]

Ahmad El Soufi, Said Ilias, and Antonio Ros, Sur la premiere valeur propre des tores, S \'e minaire de th \'e orie spectrale et g \'e om \'e trie 15 (1996), 17--23

work page 1996
[8]

Joseph Hersch, Quatre propri \'e t \'e s isop \'e rim \'e triques de membranes sph \'e riques homogenes , CR Acad. Sci. Paris Sér. AB 270 (1970), A1645--A1648

work page 1970

Show all 15 references

[9]

63, 3967--3985

Dmitry Jakobson, Michael Levitin, Nikolai Nadirashvili, Nilima Nigam, and Iosif Polterovich, How large can the first eigenvalue be on a surface of genus two?, International Mathematics Research Notices 2005 (2005), no. 63, 3967--3985

2005
[10]

2, 381--400

Dmitry Jakobson, Nikolai Nadirashvili, and Iosif Polterovich, Extremal metric for the first eigenvalue on a klein bottle, Canadian Journal of Mathematics 58 (2006), no. 2, 381--400

2006
[11]

2, 269--291

Peter Li and Shing-Tung Yau, A new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces, Inventiones mathematicae 69 (1982), no. 2, 269--291

1982
[12]

Sebasti \'a n Montiel and Antonlo Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Inventiones mathematicae 83 (1986), 153--166

1986
[13]

Nikolai Nadirashvili, Berger's isoperimetric problem and minimal immersions of surfaces, Geometric and Functional Analysis 6 (1996), no. 5, 877

1996
[14]

1, 84--98

Shin Nayatani and Toshihiro Shoda, Metrics on a closed surface of genus two which maximize the first eigenvalue of the laplacian, Comptes Rendus Mathematique 357 (2019), no. 1, 84--98

2019
[15]

102, Princeton University Press, 1982

Shing-Tung Yau, Seminar on differential geometry, no. 102, Princeton University Press, 1982

1982

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

2, 129--149

Marcel Berger, Sur les premieres valeurs propres des vari \'e t \'e s riemanniennes , Compositio Mathematica 26 (1973), no. 2, 129--149

work page 1973

[2] [2]

Robert L Bryant, On the conformal volume of 2-tori, arXiv preprint arXiv:1507.01485 (2015)

arXiv 2015

[3] [3]

4, 667--690

Donato Cianci, Mikhail Karpukhin, and Vladimir Medvedev, On branched minimal immersions of surfaces by first eigenfunctions, Annals of Global Analysis and Geometry 56 (2019), no. 4, 667--690

work page 2019

[4] [4]

Ahmad El Soufi, Hector Giacomini, and Mustapha Jazar, A unique extremal metric for the least eigenvalue of the laplacian on the klein bottle, Duke Mathematical Journal (2006)

work page 2006

[5] [5]

Ahmad El Soufi and Said Ilias, Immersions minimales, premiere valeur propre du laplacien et volume conforme, Mathematische Annalen 275 (1986), 257--267

work page 1986

[6] [6]

5, 1611--1618

, Extremal metrics for the first eigenvalue of the laplacian in a conformal class, Proceedings of the American Mathematical Society 131 (2003), no. 5, 1611--1618

work page 2003

[7] [7]

Ahmad El Soufi, Said Ilias, and Antonio Ros, Sur la premiere valeur propre des tores, S \'e minaire de th \'e orie spectrale et g \'e om \'e trie 15 (1996), 17--23

work page 1996

[8] [8]

Joseph Hersch, Quatre propri \'e t \'e s isop \'e rim \'e triques de membranes sph \'e riques homogenes , CR Acad. Sci. Paris Sér. AB 270 (1970), A1645--A1648

work page 1970