New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces

Oscar Perdomo

New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces

T0 review · 0 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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

Pith's one-line read Minimal products of three spheres admit new explicit eigenfunctions for the stability operator at eigenvalue -n, forcing the index to be at least kℓ+3k+3ℓ+8.

desk verdict Clean extension of Perdomo's own equal-dimension eigenfunctions to k eq ℓ, with a solid algebraic verification and a usable index lower bound. read the letter →

arxiv 2607.04917 v1 pith:6JZL4H6B submitted 2026-07-06 math.DG

Oscar Perdomo This is my paper

classification math.DG MSC 53C4253A10

keywords minimalhypersurfacesstabilityoperatoreigenfunctionsindexgeneralizedrotationalGaussmapspheres

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

On any compact minimal hypersurface in a sphere the coordinate functions of the Gauss map are eigenfunctions of the stability operator for the eigenvalue -n. This paper constructs a larger family of eigenfunctions, of the product form ω(t) y_i z_j, for the special minimal immersions that factor through three spheres. The weight ω is an explicit power of the two radial profile functions that define the immersion. Because these new eigenfunctions are linearly independent of the Gauss-map coordinates, the multiplicity of -n jumps, and a comparison of four reduced one-dimensional operators then yields a concrete lower bound on the stability index. The bound improves earlier estimates that treated only the equal-dimension case and applies whether or not the immersion is embedded.

What carries the argument

The reduced second-order operator S_{22} obtained by restricting the stability operator to product functions of the form η(t)y_i z_j; Theorem 2.2 shows that -n is an unexpected eigenvalue of S_{22}, which, by Rayleigh comparison with the three companion operators S_{11}, S_{21} and S_{12}, forces three further negative eigenvalues and produces the index lower bound.

What would settle it

Directly compute the second variation of a known numerical example (for instance a Carlotto–Schulz surface with small k and ℓ) and check whether the stated product functions really give eigenvalue -n and whether the counted multiplicity matches the claimed lower bound.

Watch

Extended reading notes

Core claim

For every minimal immersion ϕ(y,z,t)=(f(t)y,f₂(t)z,f₁(t)) of S^k×S^ℓ×S¹ into the sphere, the functions ζ_ij=ω(t)y_i z_j with ω=f^{-k}f₂^{-ℓ} satisfy J(ζ_ij)=-n ζ_ij. Consequently the multiplicity of the eigenvalue -n is at least (k+ℓ+3)+(k+1)(ℓ+1) and the stability index is at least kℓ+3k+3ℓ+8.

Load-bearing premise

The profile curve must satisfy the first-order ODE that encodes minimality; if that ODE fails, the algebraic cancellations that prove the new functions are eigenfunctions collapse.

Editorial extensions

If this is right

Every minimal immersion of the three-sphere product type has stability index at least kℓ+3k+3ℓ+8, independent of embedding.
The multiplicity of -n is at least (k+ℓ+3)+(k+1)(ℓ+1), strictly larger than the classical n+2 contribution coming from the Gauss map alone.
When k=ℓ the new bound recovers and extends the earlier estimate k²+6k+8.
Any further eigenfunctions or numerical spectrum computations for these immersions must account for this enlarged eigenspace before claiming completeness.

Reading between the lines

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

The same product construction may produce unexpected eigenfunctions for other eigenvalues once the profile ODE is solved, potentially tightening the index bound further.
If the long-standing conjecture that index n+3 characterises Clifford hypersurfaces is true, these examples lie strictly outside that class for every k,ℓ≥1.
The explicit weight ω suggests a recursive pattern that could generate still higher-multiplicity eigenfunctions when more spherical factors are present.

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

The new content is exactly what the title claims: for the product immersions φ(y,z,t)=(f(t)y,f_{2}(t)z,f_{1}(t)), the functions ζ_{ij}=ω(t)y_i z_j with ω=f^{-k}f_{2}^{-ℓ} are eigenfunctions of the Jacobi operator for eigenvalue -n, even when k eqℓ. That immediately gives the index lower bound kℓ+3k+3ℓ+8. The equal-dimension case was already done by the same author; this removes the restriction and writes the cancellation out in full.

The algebra is long but elementary and appears correct. After the TreadmillSled ODEs that encode minimality are substituted, every intermediate expression for Q is expanded, the n-terms cancel, the coefficient of heta' simplifies cleanly, and the remaining trigonometric polynomial vanishes by the definitions of ξ_{1},ξ_{2}. No hidden analytic assumption or circular step shows up once Lemma 2.1 and the first-order system are granted. The subsequent Rayleigh comparison of the four reduced operators S_{ij} is standard and free of gaps. Self-citation is heavy but legitimate: the Laplacian formula and the k=ℓ case are used as black boxes whose statements do not depend on the new result.

Soft spots are minor. The comparison argument that turns the unexpected eigenvalue into the index count is a bit sketchy (it relies on “at least the second/third eigenvalue” without writing the variational characterization in detail), and the paper inherits the coordinate formulae from earlier work without re-deriving them. Neither issue threatens the main identity. The result does not touch the index=n+3 conjecture for n>2, so broader impact is limited to people already working on these generalized rotational examples.

This is for specialists who track Morse index of minimal hypersurfaces in spheres, especially anyone computing or bounding indices for Carlotto–Schulz-type constructions. The math is solid enough that a serious editor should send it to referees; I would cite the bound if I needed a uniform lower estimate for this family.

Referee Report

0 major / 4 minor

Summary. The paper studies the stability operator J of compact minimal immersions φ: S^k imes S^ℓ imes S^1 o S^{k+ℓ+2} of the form (1), φ(y,z,t)=(f(t)y,f_{2}(t)z,f_{1}(t)). Theorem 2.2 asserts that the functions ζ_{ij}=ω(t) y_i z_j with ω=f^{-k} f_{2}^{-ℓ} are eigenfunctions for the eigenvalue -n (n=k+ℓ+1). The proof reduces J(ζ_{ij})=-n ζ_{ij} to the vanishing of an explicit scalar quantity Q, which is verified by substituting the arc-length TreadmillSled ODEs that encode minimality and performing a fully expanded algebraic cancellation. Corollary 2.3 then obtains the index lower bound ind(M)≥ kℓ+3k+3ℓ+8 by Rayleigh comparison of the four reduced operators S_{11}, S_{21}, S_{12}, S_{22} on the profile curve, using that -n is an unexpected eigenvalue of S_{22} and at least the second (resp. third) eigenvalue of the remaining operators.

Significance. The result supplies the first explicit family of eigenfunctions for -n beyond the Gauss-map coordinates on this class of generalized rotational minimal hypersurfaces, and it extends the author’s earlier multiplicity count from the equal-dimension case k=ℓ to arbitrary positive integers k,ℓ. The resulting index lower bound is sharp enough to be useful for comparison with known numerical values (e.g., for k=ℓ=1) and with the author’s conjectured exact index for the Carlotto–Schulz family. The derivation is elementary, fully explicit, and free of circularity once the standard Laplacian formula (Lemma 2.1) and the first-order minimality system are granted; those inputs are already established in the literature for this family. The paper therefore constitutes a clean, self-contained advance on the spectral geometry of these examples.

minor comments (4)

The title page and running header contain several typographical slips (“HYPERSURF ACES”, “UNEXPECTED MULTIPLICITY STABILITY OPERATOR”). These should be corrected for the published version.
In the introduction the author notes a typo in the formula for a_{1} in the earlier paper [8] and supplies the corrected expression a_{1}=f^{2}/h^{2}. A one-sentence reminder of the geometric meaning of a_{1} (or a brief re-derivation) would make the correction self-contained.
The final trigonometric identity that shows the remaining polynomial vanishes is expanded in full; a short remark that the same cancellation can be read as the identity | (f_{1},f_{2}) |^{2} sin^{2} heta + au =0 after substituting the definitions of ξ_{1},ξ_{2} would improve readability without lengthening the argument.
References [3] and [6] appear with arXiv identifiers dated 2026; if these are still preprints, the citation style should be made uniform with the other arXiv entries.

Circularity Check

1 steps flagged · score 1.0 of 10

Direct algebraic verification of new eigenfunctions; self-citations supply only independent black-box inputs (Laplacian formula, equal-k case).

self citation load bearing [Lemma 2.1 and the ODE system after TreadmillSled coordinates]
"We have the following expression for the Laplacian of functions on the immersion (1). Its proof can be found in [8]. … From [8] we know that the minimality of the immersion M is equivalent to the following system of differential equations: f′1=cos heta, f′2=sin heta, heta′=h^{2}/(f2 f^{2})(ℓ cos heta-n f2 ξ2)."

The Laplacian formula and the first-order minimality ODEs are imported from the author’s prior paper [8]. They are used as black-box inputs whose statements do not mention the new eigenfunctions or the index bound proved here; the subsequent algebraic cancellation that shows Q=0 is self-contained once those inputs are granted. The dependence is therefore ordinary self-citation rather than a circular reduction of the central claim.

full rationale

Theorem 2.2 is proved by substituting the arc-length TreadmillSled ODEs that encode minimality into the expression for Q and expanding; every cancellation is written out explicitly and does not presuppose the multiplicity or index claims. Lemma 2.1 (Laplacian) and the first-order system for (f1,f2, heta) are taken from the author’s earlier work [8], but those statements are independent of the new eigenfunctions heta ij= heta(t)yi zj and of the index lower bound. Corollary 2.3 then follows from ordinary Rayleigh comparison of the four reduced operators Sij once the unexpected eigenvalue of S22 is known. No fitted parameters, self-definitional normalizations, or load-bearing uniqueness theorems appear. The single self-citation chain is therefore non-circular and the score is 1.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper is pure differential geometry. It rests on the classical second-variation formula for minimal hypersurfaces in spheres, on the Laplacian formula for generalized rotational immersions proved in an earlier paper of the author, and on the first-order ODE system that characterises minimality of the profile curve. No free parameters are fitted; the only invented coordinate system (TreadmillSled) is a change of variables already used in prior work.

assumptions (3)

standard math The second-variation (Jacobi) operator of a minimal hypersurface M^n ⊂ S^{n+1} is J(f)=−Δf−n f−|A|² f, and the coordinate functions of the Gauss map are eigenfunctions with eigenvalue −n.
Classical; stated in the introduction and used throughout.
domain assumption The Laplacian on the immersion (1) is given by the formula in Lemma 2.1 (quoted from [8]).
Taken as a black box from the author’s previous paper; the present work never re-derives it.
domain assumption Minimality of (1) is equivalent to the ODE system f₁′=cos θ, f₂′=sin θ, θ′=h²/(f₂ f²)(ℓ cos θ−n f₂ ξ₂) in TreadmillSled coordinates.
Cited from [8]; every cancellation in the proof of Theorem 2.2 uses this system.

invented entities (1)

TreadmillSled coordinates (ξ₁,ξ₂) independent evidence
purpose: Rewrite the profile curve so that the minimality ODEs become first-order and the principal curvatures admit simple algebraic expressions.
Introduced by the author in earlier papers on helicoidal surfaces; used here as a computational device, not a new geometric object.

how reviews work

0 comments

Cite this review

Pith. "Pith review of New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces." pith.science (2026). https://pith.science/paper/6JZL4H6B

@misc{pith2026260704917,
  author       = {Pith},
  title        = {Pith review of: New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JZL4H6B}},
  note         = {Machine review of arXiv:2607.04917}
}

abstract

For any $n$-dimensional compact minimal hypersurface of the $(n+1)$-dimensional sphere, we have that the coordinate functions of the Gauss map are eigenfunctions of the stability operator associated with the eigenvalue $-n$. In this paper we consider minimal immersions from $\mbS^{k}\times\mbS^{\ell}\times\mbS^{1}$ to $\mbS^{k+\ell+2}$ of the form $\phi(y,z,t)=\left(f(t) y, f_2(t) z, f_1(t)\right)$ and we explicitly show new eigenfunctions for the stability operator associated with the same eigenvalue. We also show that the stability index of these minimal immersions is at least $k\ell+3k+3\ell+8$.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

17 extracted references · 4 linked inside Pith

[1]

8, 33 pp

Carlotto, A., and Schulz, M.Minimal hypertori in the four-dimensional sphere, Ars Inven Anal, (2023), Paper No. 8, 33 pp

2023
[2]

B., and Wiygul, D.Index growth not imputable to topology, Proc

Carlotto, A., Schulz, M. B., and Wiygul, D.Index growth not imputable to topology, Proc. Amer. Math. Soc.153(2025), no. 4, 1787-1801

2025
[3]

and Tsiamis, R.New minimal surfaces in the sphere from capillarity minimal cones, arXiv:2602.20124v1 (2026)

Firester, B. and Tsiamis, R.New minimal surfaces in the sphere from capillarity minimal cones, arXiv:2602.20124v1 (2026)

arXiv 2026
[4]

and Wei, G.New examples of constant mean curvature hypersurfaces in the sphere, arXiv:2209.13236v1 (2025)

Huang, C. and Wei, G.New examples of constant mean curvature hypersurfaces in the sphere, arXiv:2209.13236v1 (2025)

arXiv 2025
[5]

and Wei, G.Embedded constant mean curvature hypertori in the2n-sphere, Nonlinearity,38 (2025), no

Lai, J. and Wei, G.Embedded constant mean curvature hypertori in the2n-sphere, Nonlinearity,38 (2025), no. 8, Paper No. 085003 , 11 pp

2025
[6]

and Wei, G.Minimal hypersurfaces in spheres generated by isoparametric foliations, arXiv:2603.0367v1 (2026)

Lai, J. and Wei, G.Minimal hypersurfaces in spheres generated by isoparametric foliations, arXiv:2603.0367v1 (2026)

arXiv 2026
[7]

and Neves, A.Min-max theory and the Willmore conjecture, Ann

Marques, F. and Neves, A.Min-max theory and the Willmore conjecture, Ann. of Math. (2)179(2014), n. 2, 683-782

2014
[8]

Perdomo, O.Spectrum of the Laplacian and the Jacobi operator on generalized rotational minimal hypersurfaces of spheres, J. Geom. Anal.35(2025), n. 12, Paper No. 390, 25 pp

2025

Show all 17 references

[9]

Perdomo, O.The stability index and Yau’s conjecture for Carlotto-Schulz minimal hypertori, arXiv:2508.09104 (2025)

2025
[10]

Perdomo, O.The stability index and Yau’s conjecture for Carlotto-Schulz minimal hypertori, part II, arXiv:2606.16980 (2026)

2026
[11]

Perdomo, O.Existence of a constant-mean-curvature hypersurfaces inS 4 via computer assistance, arXiv:2506.19555 (2025)

2025 arXiv
[12]

Perdomo, O.Navigating the space of compact CMC hypersurfaces in spheres, part I, arXiv:2503.13823 (2025)

2025 arXiv
[13]

Perdomo, O.Navigating the space of compact CMC hypersurfaces in spheres, part II, arXiv:2505.09102 (2025)

2025 arXiv
[14]

Perdomo, O.The round Taylor method, Amer. Math. Monthly126, (2019) no. 3, pp 237-251

2019
[15]

Math.57(1) (2013), pp 87-104

Perdomo, O.Helicoidal minimal surfaces inR3, Illinois J. Math.57(1) (2013), pp 87-104

2013
[16]

Pacific J

Perdomo, O.A dynamical interpretation of the profile curve of CMC twizzler surfaces. Pacific J. Math. 258(2) (2012), pp 459-485

2012
[17]

Urbano, F.Minimal surfaces with low index in the three dimensional sphere, Proc. Amer. Math. Soc. 108(1990), n. 4, 989-992. Central Connecticut State University Email address:perdomoosm@ccsu.edu

1990

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

8, 33 pp

Carlotto, A., and Schulz, M.Minimal hypertori in the four-dimensional sphere, Ars Inven Anal, (2023), Paper No. 8, 33 pp

2023

[2] [2]

B., and Wiygul, D.Index growth not imputable to topology, Proc

Carlotto, A., Schulz, M. B., and Wiygul, D.Index growth not imputable to topology, Proc. Amer. Math. Soc.153(2025), no. 4, 1787-1801

2025

[3] [3]

and Tsiamis, R.New minimal surfaces in the sphere from capillarity minimal cones, arXiv:2602.20124v1 (2026)

Firester, B. and Tsiamis, R.New minimal surfaces in the sphere from capillarity minimal cones, arXiv:2602.20124v1 (2026)

arXiv 2026

[4] [4]

and Wei, G.New examples of constant mean curvature hypersurfaces in the sphere, arXiv:2209.13236v1 (2025)

Huang, C. and Wei, G.New examples of constant mean curvature hypersurfaces in the sphere, arXiv:2209.13236v1 (2025)

arXiv 2025

[5] [5]

and Wei, G.Embedded constant mean curvature hypertori in the2n-sphere, Nonlinearity,38 (2025), no

Lai, J. and Wei, G.Embedded constant mean curvature hypertori in the2n-sphere, Nonlinearity,38 (2025), no. 8, Paper No. 085003 , 11 pp

2025

[6] [6]

and Wei, G.Minimal hypersurfaces in spheres generated by isoparametric foliations, arXiv:2603.0367v1 (2026)

Lai, J. and Wei, G.Minimal hypersurfaces in spheres generated by isoparametric foliations, arXiv:2603.0367v1 (2026)

arXiv 2026

[7] [7]

and Neves, A.Min-max theory and the Willmore conjecture, Ann

Marques, F. and Neves, A.Min-max theory and the Willmore conjecture, Ann. of Math. (2)179(2014), n. 2, 683-782

2014

[8] [8]

Perdomo, O.Spectrum of the Laplacian and the Jacobi operator on generalized rotational minimal hypersurfaces of spheres, J. Geom. Anal.35(2025), n. 12, Paper No. 390, 25 pp

2025