Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

Daniel Stern; Mikhail Karpukhin; Peter McGrath

Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

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Pith's one-line read Within the class of minimal surfaces obtained by maximizing the first Laplace eigenvalue under basic reflection symmetries, the Lawson surfaces $\xi_{\gamma,1}$ are the least-area surfaces of their genus once the genus is large, and all…

desk verdict Karpukhin–McGrath–Stern confirm Kusner and Kapouleas conjectures in the extremal class and identify the B^3 analog of Lawson surfaces; the analytic core is delicate but plausible. read the letter →

arxiv 2502.10225 v1 pith:2WQ4YEV3 submitted 2025-02-14 math.DG math.SP

Mikhail Karpukhin , Peter McGrath , Daniel Stern This is my paper

classification math.DGmath.SP MSC 53C4258J50

keywords minimalsurfacesLaplaceeigenvaluesSteklovLawsonxi_gamma_1freeboundaryspectralextremalmetricsvarifoldconvergencelargegenusasymptotics

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 studies the geometry of the minimal surfaces in $\mathbb{S}^3$ and free boundary minimal surfaces in $\mathbb{B}^3$ produced by maximizing the first Laplace or Steklov eigenvalue among "basic reflection" symmetry classes. Its central claim is that, once the genus is large, Lawson's classical surfaces $\xi_{\gamma,1}$ have the least area among all these spectrally extremal genus-$\gamma$ surfaces, while the generic families have area exponentially close to $8\pi$ and therefore converge to a great sphere counted twice. In the ball, the analogous Steklov results control the area gap in terms of genus and boundary components, and single out a two-plane-symmetric family whose area gap decays like $1/\gamma$, the free-boundary analog of the Lawson family. The net effect is a sharp quantitative picture of low-area minimal surfaces: the Lawson family sits at the bottom of the possible areas, and every other family produced by the construction disappears into a double equator.

What carries the argument

The load-bearing reduction is that every basic reflection pair can be replaced by a genus-zero domain $\Omega \subset \mathbb{S}^2$ (or $\Omega \subset \mathbb{D}^2$ in the Steklov case) on which the optimization problem becomes the supremum of area times $\min\{\lambda_1^D, \lambda_1^N\}$ (or boundary length times $\min\{\sigma_1^D, \sigma_1^N\}$); doubling the domain recovers the original surface, so $\Lambda_1(M,\Gamma) = 2 \sup \bar\mu_1(\Omega,g)$. Lower bounds are built from explicit domains, spheres with small disks removed, whose first Dirichlet eigenvalue is at least $2$ and whose total removed area is $e^{-c\sqrt{k}}$ or $e^{-ck}$, combined with a Neumann estimate showing $\lambda_1^N |\Omega| \geq 8\pi - C|D|$. Upper bounds use stability: a small area gap forces the first-eigenfunction embedding to be close to the standard inclusion of $\mathbb{S}^2$, and a uniform spectral gap $\lambda_4^N - 2 > c$, proved by a mean-value $L^\infty$ estimate and a compactness argument passing eigenfunctions to the round sphere, upgrades this to exponential decay of the distance to an equator. The Steklov half runs the same two-step program with mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues on domains in the disk.

What would settle it

Construct a sequence of genus-zero free boundary minimal surfaces $N_k \subset \mathbb{S}^3_+$ embedded by first eigenfunctions with $4\pi - |N_k| = o(1/k)$ whose varifold limit is not the equator; Theorem 1.9 and Theorem 1.18 would be false. Equivalently, compute the fourth Neumann eigenvalue on the halves of such surfaces with gap $\leq \varepsilon/k$: if $\liminf (\lambda_4^N - 2) = 0$, Lemma 6.2 is false and the exponential closeness conclusions do not follow.

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Extended reading notes

Core claim

The paper's central theorem pair is this: for every orientable basic reflection pair $(M,\Gamma)$ of genus $\gamma \geq \gamma_0$, the extremal first Laplace eigenvalue satisfies $\tfrac{1}{2}\Lambda_1(M,\Gamma) \geq \mathrm{Area}(\xi_{\gamma,1})$, and for generic pairs the stronger bound $\tfrac{1}{2}\Lambda_1(M,\Gamma) \geq 8\pi - e^{-c\sqrt{\gamma}}$ holds. Complementing this, Theorem 1.5 gives the universal upper bound $\tfrac{1}{2}\Lambda_1(M,\Gamma) \leq 8\pi - e^{-C_1|\chi(M)|}$, improved to $8\pi - C_2/|\chi(M)|$ for Scherk-type pairs. Together these bounds imply that as genus tends to infinity every non-Lawson family in this class converges as a varifold to a multiplicity-two great sphere, while the Lawson family is singled out as the desingularization of a pair of orthogonal great spheres. In the Steklov setting, Theorems 1.12, 1.13, 1.15, and 1.18 show that the area gap $4\pi - \Sigma_1(N,\Gamma)$ is exponentially small in the number of boundary components, can decay as slowly as $1/\gamma$ in genus for a two-plane-symmetric family, and forces rapid varifold convergence to an equatorial disk whenever the gap is $o(1/(1+|\chi(N)|))$.

Load-bearing premise

The load-bearing premise is Lemma 6.2 and its Steklov analogue Lemma 7.3: whenever the area gap is at most $\varepsilon/k$, the fourth Neumann eigenvalue stays strictly above $2$ by a positive constant, and if the mean-value $L^\infty$ estimate or the passage of eigenfunctions to the round sphere fails, the generic exponential-closeness conclusions and the Scherk-type upper bound collapse.

Editorial extensions

If this is right

For every basic reflection pair of genus $\gamma \geq \gamma_0$, the Lawson surface $\xi_{\gamma,1}$ has area no larger than the spectrally extremal surface, so the least-area conjecture holds across this whole class at large genus.
Generic pairs satisfy $\tfrac{1}{2}\Lambda_1(M,\Gamma) \geq 8\pi - e^{-c\sqrt{\gamma}}$ and therefore converge in the varifold sense to a great sphere of multiplicity two as $\gamma \to \infty$; they cannot converge to a pair of orthogonal spheres.
Every basic reflection surface obeys the universal upper bound $\tfrac{1}{2}\Lambda_1(M,\Gamma) \leq 8\pi - e^{-C|\chi(M)|}$, so the area gap can never be smaller than exponentially small in the Euler characteristic.
Scherk-type pairs satisfy the polynomial bound $\tfrac{1}{2}\Lambda_1(M,\Gamma) \leq 8\pi - C_2/|\chi(M)|$, which is of the same order as the known area deficit of the Lawson surfaces; the paper conjectures equality in these cases.
In $\mathbb{B}^3$, the Steklov extremal free boundary minimal surfaces have area gap controlled by $\max\{e^{-c'k}, C/\gamma\}$, and the two-plane-symmetric family realizes the slow $1/\gamma$ decay, matching the role of the Lawson surfaces.

Reading between the lines

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

Beyond the paper: the Scherk-type/generic dichotomy suggests that for any equivariant eigenvalue maximizer, the varifold limit is either an orthogonal pair of spheres or a multiplicity-two equator; the same stability argument could be run on reflection groups not covered here to test this.
Beyond the paper: the $e^{-c\sqrt{\gamma}}$ lower bound for generic pairs may not be sharp, since several constructions in Section 4 give $e^{-c\gamma}$ decay; a finer computation could separate the generic families into two exponential rates.
Beyond the paper: the two-plane-symmetric free boundary family is the natural free-boundary stand-in for the Lawson surfaces; comparing it with the larger three-reflection group would decide whether adding symmetry lowers the extremal area, as the numerical evidence cited in the paper suggests.
Beyond the paper: if the conjectured equality for Scherk-type pairs holds, the spectral optimization problem itself becomes a variational construction of the Lawson surfaces, giving an independent proof of their uniqueness among first-eigenfunction embeddings with those symmetries.

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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

Bottom line: this is a strong paper, and the reader's ACCEPT is right. The genuinely new content is the Scherk-type/generic dichotomy, the exponential vs 1/|χ| area asymptotics for the Λ1-extremal basic reflection surfaces, the varifold convergence to a multiplicity-two great sphere for generic surfaces, and the identification of the Γ12 free-boundary family with 1/γ area gap. If the main results hold—and I think they do—they confirm Kusner's least-area conjecture and Kapouleas's dichotomy conjecture within this large class, and give a clean free-boundary analog. That is a real advance, not a repackaging.

What the paper does well: the lower bounds are genuine constructions. Test metrics with controlled Dirichlet eigenvalues (Lemmas 4.12, 4.14, 4.16) and the stability upper bounds are the right machinery, and the reduction to shape optimization over domains in S^2 and D^2 (Lemmas 2.1, 2.2) makes the problem tractable. The paper is honest about what depends on [KKMS24] and about what remains conjectural (Conjectures 1.7, 1.17). The citation pattern is fine; the reliance on their earlier existence theory is stated, and Choe–Soret's λ1(ξ)=2 is an independent input.

Soft spots, in proportion: the delicate core is Lemma 6.2 (and its Steklov analogue Lemma 7.3), the uniform spectral gap λ4−2>c under δ ≤ ε/k. The proof is sketched in places—the compactness passage, the C^1 estimate for low-frequency projections, and the diagonal argument for j=1..4 are not fully written out. I looked for a concrete failure; the mean-value estimate (6.4) is standard Colding–Minicozzi, the W^{1,1} compactness on S^2 uses ‖dρ_k‖_{L^1}≤C√(kδ_k) and |∂Ω_k|≤C√(kδ_k), both o(1) under δ≤ε/k, and the limit passage to round-sphere eigenfunctions checks out. This is ordinary analytic bookkeeping, not a discovered gap. Proposition 5.6 is similar: the proof is abbreviated, but the only real assumption is the gap σ3^D≥1+δ0, which the preceding argument establishes. No load-bearing flaw found.

Who is this for: anyone working on spectral extremal metrics, minimal surfaces in S^3/B^3, or doublings. It deserves a serious referee—specialist in eigenvalue optimization or minimal surface regularity—and the referee should be asked to check Lemma 6.2, Lemma 7.3, and Proposition 5.6 in detail. My own verdict is accept after that check; the structural claims and the new examples are solid enough that a desk rejection would be wrong.

Referee Report

0 major / 6 minor

Summary. The paper studies the large-topology behavior of minimal surfaces in S^3 and free boundary minimal surfaces in B^3 obtained in the authors' earlier work with Kusner via equivariant optimization of the first Laplace and Steklov eigenvalues. For basic reflection pairs (M,Γ), the main results are: (i) for all sufficiently large genus, (1/2)Λ_1(M,Γ) ≥ Area(ξ_{γ,1}), with the generic cases satisfying (1/2)Λ_1 ≥ 8π − e^{-c√γ}; (ii) universal upper bounds (1/2)Λ_1 ≤ 8π − e^{-C|χ(M)|}, improved to 8π − C_2/|χ(M)| for Scherk-type pairs; (iii) a rapid varifold convergence statement for generic pairs, showing convergence to a multiplicity-two great sphere; and (iv) analogous Steklov estimates in B^3, including a family whose area gap is of order 1/|χ| and which the authors propose as the free boundary counterpart of the Lawson surfaces. The proofs combine explicit test-metric constructions with Dirichlet and Neumann eigenvalue estimates on domains in S^2 and D^2, stability arguments in the spirit of Karpukhin–Nahon–Polterovich–Stern and Karpukhin–Stern, and compactness/mean-value arguments for eigenfunctions.

Significance. If the results stand, they constitute a substantial step toward two well-known conjectures, Kusner's least-area conjecture for Lawson surfaces and Kapouleas's question on possible varifold limits of low-area minimal surfaces in S^3, at least for the large class of surfaces arising from equivariant eigenvalue optimization. The paper also gives the first sharp asymptotic dichotomy, exponential versus polynomial area gaps, among the constructed families, and identifies a plausible free boundary analogue of the ξ_{γ,1} surfaces. The lower bounds come from explicit test metrics with universal constants and no fitted parameters, and the upper bounds are derived from stability inequalities; the main analytic core, Lemma 6.2 and its Steklov analogue, is supplied with a proof. The paper is not self-contained: it relies substantially on the existence theory of [KKMS24] and on the Choe–Soret computation λ_1(ξ_{γ,1})=2, but these are legitimate independent inputs. The claims are precise and falsifiable, and the paper is a strong contribution to spectral geometry and minimal surface theory.

minor comments (6)

[Title] The header of the paper reads 'SPECTRALL Y' and should be corrected to 'SPECTRALLY'.
[Section 4.1] The corollary labelled 'Corollary 4.4' appears after Proposition 4.10; this numbering is out of sequence and should be corrected (for example by renumbering it as Corollary 4.11 or adjusting the intervening labels).
[Section 6.1, proof of Theorem 1.9] There is a typo in the phrase 'Prposition 6.3', which should read 'Proposition 6.3'.
[Section 6.1, proof of Lemma 6.2] In the boundary-term estimate near the end of the proof, the displayed inequality '|∂Ω_k| ≤ C∑ r_i ≤ C√(kε_k) < C√δ_k' is inconsistent with the hypothesis δ_k ≤ ε_k/k; the intended bound appears to be '≤ C√(kδ_k) ≤ C√ε_k', and the displayed estimate should be corrected accordingly.
[Section 7.1, proof of Lemma 7.3] Lemma 7.4 is stated in the middle of the proof of Lemma 7.3 and then used in that proof; stating the auxiliary estimate before the proof of Lemma 7.3 would improve readability, since it is a separate input to the argument.
[Proof of Theorem 1.5 completed] The application of Theorem 1.9 to a closed surface M is correct only after passing to the two halves of M in S^3_+; this passage is implicit and should be stated explicitly, since Theorem 1.9 is formulated for free boundary minimal surfaces.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the area bounds are derived from explicit test metrics and independent stability estimates, not from fitted constants or self-citation chains.

full rationale

The paper's lower bounds for Λ1 and Σ1 come from direct constructions of test domains in S² and D, with eigenvalue estimates proven in Sections 3–5. The upper bounds come from stability arguments in Sections 6–7, centered on Lemmas 6.2 and 7.3, whose proofs use mean-value estimates, compactness, and W^{1,1} convergence to round-sphere eigenfunctions; they do not assume the conclusions they prove. The Scherk-type lower bound in Lemma 4.7 uses an explicit diffeomorphism to the Lawson surface and the independent eigenvalue computation λ1(ξγ,1)=2 of Choe–Soret [CS09], which is external evidence and not a restatement of the present results. The existence of the extremal surfaces is imported from the authors' prior paper [KKMS24], but that existence theorem is parameter-free, does not assume the area estimates proven here, and is used only as an input: the present paper proves new geometric bounds about those surfaces. Citations to [KNPS21, KS24] import stability techniques, not conclusions. No equation is defined in terms of a target quantity, and no fitted parameter is renamed as a prediction. The comparison with Area(ξγ,1) uses the independent asymptotics of [HHT23, CHHT24]. Overall, the derivation chain is self-contained given the stated existence theory, and no circular step is present.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No data fitting or empirical constants appear. The universal constants C, c, ε0, γ0 are existential constants chosen in the proofs; their values do not encode the conclusions. The surfaces are defined by variational problems, not by fitting parameters. No new physical or geometric objects are postulated beyond the surfaces already defined in KKMS24.

assumptions (8)

standard math Uniformization theorem for compact surfaces with boundary: any metric on a genus-zero surface with boundary is conformal to S^2 minus a union of geodesic disks.
Used in Section 2 (Lemma 2.1) and Lemma 4.7 to pass from basic reflection pairs to domains in S^2.
standard math Classification of finite reflection groups on S^2 (tetrahedral, octahedral, icosahedral, Z2 × Dk, Dk, Z2, trivial).
Used in Section 4.1 to enumerate Scherk-type and generic types for the lower-bound constructions.
standard math Choe-Soret theorem: the first Laplace eigenvalue of Lawson surfaces ξ_{γ,1} is 2.
Used in Lemma 4.7 to obtain Λ_1(M,Γ) ≥ 2Area(ξ_{γ,1}) and in Conjecture 1.7.
standard math Existence of λ̄_1-maximizing metrics and associated minimal embeddings for orientable basic reflection pairs (M,Γ) with Γ=⟨τ⟩×G (KKMS24, Theorem 8.7).
The paper's objects of study are these extremal surfaces; Theorem 1.1 is taken from the authors' prior work.
domain assumption Existence of σ̄_1-maximizing metrics for a large class of basic reflection pairs with boundary (KKMS24, Theorem 9.15), plus an announced extension by Petrides to all basic reflection pairs.
Theorems 1.12-1.15 and 1.18 concern these free boundary surfaces; the full existence for every pair is not yet established (Remark 1.11).
standard math Montiel-Ros theorem: a conformal diffeomorphism of a minimal surface embedded by first eigenfunctions is an isometry.
Used in the proof of Theorem 1.5 to extend the symmetry group of the extremal metric.
standard math Osgood-Phillips-Sarnak theorem: any metric on a surface with boundary is conformal to one with totally geodesic boundary.
Used in Lemma 2.1 to relate metrics on fundamental domains to those on the double.
standard math Stability estimates for isoperimetric eigenvalue inequalities from Karpukhin-Nahon-Polterovich-Stern and Karpukhin-Stern.
Used in Sections 6 and 7 for the upper bounds and the convergence-to-equator results.

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Pith. "Pith review of Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$." pith.science (2026). https://pith.science/paper/2WQ4YEV3

@misc{pith2026250210225,
  author       = {Pith},
  title        = {Pith review of: Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbbB^3$ and $\mathbbS^3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WQ4YEV3}},
  note         = {Machine review of arXiv:2502.10225}
}

abstract

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in $\mathbb{S}^3$ with area below $8\pi$, and many new free boundary minimal embeddings in $\mathbb{B}^3$ with area below $2\pi$. In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in $\mathbb{S}^3$ in this class of examples and the special role played by Lawson's $\xi_{\gamma,1}$ surfaces. We also confirm analogous statements in $\mathbb{B}^3$ and identify a family of free boundary minimal surfaces in $\mathbb{B}^3$ most closely resembling $\xi_{\gamma,1}$.

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Pith tools

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

[1] [1]

Conway, Heidi Burgiel, and Chaim Goodman-Strauss

John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. The symmetries of things . A K Peters, Ltd., Wellesley, MA, 2008

work page 2008

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Minimal surfaces and alternating multiple zetas

Steven Charlton, Lynn Heller, Sebastian Heller, and Martin Traizet. Minimal surfaces and alternating multiple zetas. arXiv preprint arXiv:2407.07130 , 2024

arXiv 2024

[3] [3]

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Tobias Holck Colding and William P. Minicozzi, II. A course in minimal surfaces , volume 121 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2011

work page 2011

[4] [4]

First eigenvalue of symmetric minimal surfaces in S^3

Jaigyoung Choe and Marc Soret. First eigenvalue of symmetric minimal surfaces in S^3 . Indiana Univ. Math. J. , 58(1):269--281, 2009

work page 2009

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Giada Franz, Daniel Ketover, and Mario B Schulz. Genus one critical catenoid. arXiv preprint arXiv:2409.12588 , 2024

arXiv 2024

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Free boundary minimal surfaces in the unit 3-ball

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Giada Franz and Mario Schulz. Topological control for min-max free boundary minimal surfaces. arXiv preprint arXiv:2307.00941 , 2023

arXiv 2023