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Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

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Pith's one-line read The paper proves sharp upper bounds on the second eigenvalue of the Jacobi operator for surfaces with nonpositive Euler characteristic in spheres and in circle-times-sphere products, and classifies the equality cases as the Clifford…

desk verdict Sharp second-eigenvalue rigidity for surfaces in spheres and a product space, with one repairable gap in the equality case of Theorem B. read the letter →

arxiv 2505.22439 v1 pith:XXGDLHI6 submitted 2025-05-28 math.DG

Márcio Batista , Marcos P. Cavalcante , Abraão Mendes , Ivaldo Nunes This is my paper
classification math.DG MSC 53C2458J5049Q0553A10
keywords JacobioperatorsecondeigenvaluespectralrigidityWillmoreenergyminimalsurfacesCliffordtorusequilateralproductspaces
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

Closed surfaces with nonpositive Euler characteristic cannot have an arbitrarily large second eigenvalue of the Jacobi operator: in the unit sphere the bound is $\lambda_2(L) \le -2$, and in the product $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$ with $r \ge s$ the bound is $\lambda_2(L) \le 0$. The paper proves these inequalities and then asks when equality is possible. Equality in the sphere forces the surface to be orientable and to be either the Clifford torus in $\mathbb{S}^3$ or the equilateral torus in $\mathbb{S}^5$. Equality in the product forces the two factor radii to be equal and the surface to be the totally geodesic torus $\mathbb{S}^1(r) \times \mathbb{S}^1(r)$. The interest is that a single spectral quantity, the second eigenvalue, recognizes these classical minimal tori.

What carries the argument

The engine is the variational characterization of $\lambda_2$ together with conformal test functions. Taking a positive first eigenfunction $\varphi$ of $L$, the paper follows the conformal-centre argument of [18] to choose $y \in B^{n+1}$ such that the coordinate functions $\psi_i$ of $F_y \circ x$ are orthogonal to $\varphi$; these $\psi_i$ are then admissible in the Rayleigh quotient for $\lambda_2$. The Dirichlet energy of $\psi$ is twice the area of the transformed surface, and Lemma 2.1 bounds that area by the Willmore energy. In the product-space setting, Proposition 2.2 bounds the ambient contribution $|\rho|^2_\Sigma$ by $2r^2/(1-r^2)$, and the remaining computation converts the inequalities into $\lambda_2|\Sigma| \le 4\pi\chi - 2\int |H|^2$. The named objects are the Jacobi operator and the conformal transformations $F_y$ of the sphere.

What would settle it

A decisive calculation is to find a minimal torus in $\mathbb{S}^1(1/\sqrt{2}) \times \mathbb{S}^2(1/\sqrt{2})$ that satisfies the equality condition of Proposition 2.2, namely $2K = -|\sigma|^2$, and has $|\sigma|$ not identically zero; if such a torus exists and $\lambda_2(L) = 0$, then Theorem B's rigidity classification is false, while a proof that minimality plus $2K = -|\sigma|^2$ forces $K = 0$ would close the gap in the proof.

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Extended reading notes

Core claim

Let $\Sigma$ be a closed surface fully immersed in the unit sphere $\mathbb{S}^n$ and let $L = -\Delta - |\sigma|^2 - 2$ be its Jacobi operator. The paper proves $\lambda_2(L)|\Sigma| \le -2W(\Sigma) + 4\pi\chi(\Sigma)$, where $W$ is the Willmore energy and $\chi$ the Euler characteristic; consequently $\lambda_2(L) \le -2$ whenever $\chi(\Sigma) \le 0$. Equality $\lambda_2(L) = -2$ forces minimality, flatness, and a first-eigenfunction immersion, and the classification of tori and Klein bottles then leaves exactly the Clifford torus in $\mathbb{S}^3$ and the equilateral torus in $\mathbb{S}^5$. For a closed orientable surface of positive genus immersed in $M = \mathbb{S}^1(r) \times \mathbb{S}^2(s)$ with $r \ge s$, the paper proves $\lambda_2(L) \le 0$ for the Jacobi operator $L = -\Delta - (|\sigma|^2 + \mathrm{Ric}_M(N,N))$; equality forces $r = s$ and congruency to the totally geodesic torus $\mathbb{S}^1(r) \times \mathbb{S}^1(r)$. The proof uses Proposition 2.2 to show that the equality case with $r = 1/\sqrt{2}$ forces the surface to be flat and totally geodesic.

Load-bearing premise

Theorem B's rigidity conclusion depends on the step in which the equality case of Proposition 2.2, together with minimality, is said to force the surface to be flat and totally geodesic; the equations written only give $2K = -|\sigma|^2$ under minimality, which does not by itself imply $K = 0$ and $|\sigma| = 0$.

Editorial extensions

If this is right

  • For any closed surface with $\chi(\Sigma) \le 0$ fully immersed in $\mathbb{S}^n$, the second Jacobi eigenvalue is at most $-2$, and if equality holds the surface must be one of the two classical minimal tori.
  • In $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$ with $r \ge s$, every orientable positive-genus immersed surface has $\lambda_2(L) \le 0$; the only way to reach $0$ is $r = s$ and the totally geodesic torus.
  • Together with the solution of the Willmore conjecture, the sphere bound yields $\lambda_2(L)|\Sigma| \le -4\pi^2$ for orientable positive-genus surfaces in $\mathbb{S}^3$, with equality only for the Clifford torus up to conformal transformations.
  • The product-space theorem supplies an alternative proof of the index-one rigidity of the totally geodesic torus in $\mathbb{S}^1(r) \times \mathbb{S}^2(r)$.
  • The sharp inequality $\lambda_2(L)|\Sigma| \le -2W(\Sigma) + 4\pi\chi(\Sigma)$ holds uniformly for all fully immersed closed surfaces in $\mathbb{S}^n$, giving a spectral route to Willmore-type area control.

Reading between the lines

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

  • The same conformal test-function scheme should give sharp $\lambda_2$ bounds in other ambients with a transitive conformal group, such as hyperbolic space, once a Willmore-type area estimate is available.
  • A testable extension is to compute $\lambda_2(L)$ for the bipolar Klein bottle in $\mathbb{S}^4$; Theorem A predicts the strict inequality $\lambda_2 < -2$ for it, so such a computation would check the sharpness of the bound outside dimensions 3 and 5.
  • The product example in Section 5 with $r < s$ shows that equality $\lambda_2 = 0$ can occur away from the equality classification, which suggests that the role of the hypothesis $r \ge s$ is to exclude those flat tori; the same exclusion mechanism might be adapted to other warped-product ambients.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the second eigenvalue of the Jacobi operator L = -Δ - |σ|² - 2 for closed immersed surfaces with nonpositive Euler characteristic in the unit sphere, and L = -Δ - (|σ|² + Ric_M(N,N)) for surfaces in the product space S¹(r)×S²(s). Theorem A states that for a closed surface Σ fully immersed in S^n with χ(Σ) ≤ 0 one has λ₂(L) ≤ -2, with equality exactly when Σ is the Clifford torus in S³ or the equilateral torus in S⁵. Theorem B states that for a closed orientable positive-genus surface immersed in S¹(r)×S²(s) with r ≥ s one has λ₂(L) ≤ 0, and equality forces r = s and Σ to be the totally geodesic torus S¹(r)×S¹(r). The proofs use Li–Yau conformal-area techniques, the Willmore energy, and external classification theorems of El Soufi–Ilias and El Soufi–Giacomini–Jazar, with a final section giving explicit flat-torus examples showing the necessity of the hypothesis r ≥ s.

Significance. If the results are correct, they provide sharp spectral rigidity statements for the Jacobi operator in two natural ambient settings, extending earlier work of Harrell–Loss and El Soufi–Ilias. The main estimates are clean and essentially parameter-free: Lemma 2.1 gives a sharp conformal-area bound, Theorem 3.3 gives a Willmore-energy estimate that immediately implies the eigenvalue bound in Theorem A, and the inequality half of Theorem B follows by a coherent chain of inequalities. The paper is honest about relying on external classifications for the equality cases, and the explicit examples in Section 5 usefully demonstrate that the hypothesis r ≥ s in Theorem B cannot simply be dropped. The main weakness is a localized but load-bearing gap in the rigidity half of Theorem B, where the passage from equality in Proposition 2.2 to flatness and total geodesy is asserted without the needed Gauss–Bonnet argument; this gap is readily repairable.

major comments (2)
  1. [Section 4, proof of Theorem 4.1, item (iii)] The sentence 'By Proposition 2.2, equality in (iii) implies r = 1/√2 and that Σ is flat and totally geodesic in M' is not a direct consequence of Proposition 2.2 as stated. In the relevant case r = 1/√2, the equality case (ii) of Proposition 2.2 gives either a slice {θ}×S²(1/√2), which is incompatible with χ(Σ) = 0, or the pointwise relation 2K = 4|H|² - |σ|². With minimality H = 0 this gives 2K = -|σ|², which by itself does not force K = 0 or |σ| = 0. The missing step is to combine 2K = -|σ|² ≤ 0 with Gauss–Bonnet and χ(Σ) = 0 to conclude K ≡ 0 and |σ| ≡ 0, and then to rule out the slice alternative explicitly; this argument should be inserted before the rigidity conclusion is drawn.
  2. [Section 4, proof of Theorem 4.1, final paragraph] The final sentence says 'Since L = ∆+2 and λ2(L) = 0, it follows that Σ must coincide with this torus.' From the definition of L in Theorem B, on a totally geodesic torus one has L = -Δ - 2, so the displayed sign appears to be a typo. More importantly, the step from 'Σ is a finite covering of the totally geodesic torus S¹(1/√2)×S¹(1/√2)' to 'Σ coincides with it' needs a justification: a proper finite covering has first nonzero Laplacian eigenvalue strictly smaller than 2, hence λ₂(L) < 0, so equality λ₂(L) = 0 forces the covering to be trivial. This one-sentence argument should be written out.
minor comments (5)
  1. [Title and abstract] The title and abstract contain typographical artifacts, such as 'SURF ACES' and 'V ALUE', which should be cleaned in the published version.
  2. [Section 2.1, Lemma 2.1] The notation z_N is used without a formal definition; a sentence clarifying that it denotes the component of z normal to Σ and tangent to S^n would improve readability.
  3. [References] Reference [13] lists the third author as 'Stern Daniel'; the conventional ordering is 'Daniel Stern'.
  4. [Section 5, examples] The cases in the examples are phrased as r ≥ t and r ≤ t, while the theorem's hypothesis is r ≥ s; since s² = t² + h², the relation between t and s is clear, but restating it explicitly would help the reader connect the examples to the theorem.
  5. [Section 4, proof of Theorem 4.1] The eigenvalue computation for the totally geodesic torus would be clearer if written out: μ₁(-Δ) = 2 on S¹(1/√2)×S¹(1/√2), and therefore λ₂(L) = μ₁(-Δ) - 2 = 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained modulo standard geometric identities and external classification theorems; the only flagged issue is a fillable proof gap, not a circular step.

full rationale

The derivation chain is not circular. In Theorem A (Section 3), the second eigenvalue bound is obtained by using the first eigenfunction to select a conformal map via the Li-Yau argument, then testing the second eigenvalue with the coordinate functions of the conformally transformed immersion. The ingredients are the Gauss equation, the Gauss-Bonnet theorem, the Willmore-area estimate of Lemma 2.1, and the external classification theorems of El Soufi-Ilias [6] and El Soufi-Giacomini-Jazar [4]. No parameter is fitted to the target equality, and no target conclusion is assumed in the test-function estimate. In Theorem B (Section 4), the estimate again uses the Li-Yau test functions, Lemma 2.1, the Gauss equation, and the independently proved bound in Proposition 2.2; equality yields H=0, chi=0, and equality in Proposition 2.2. The only problematic sentence is in the proof of Theorem 4.1, item (iii): the manuscript states that equality in Proposition 2.2 implies r=1/sqrt(2) and flatness/total geodesicity, but Proposition 2.2 alone gives, in the relevant case, 2K=4|H|^2-|sigma|^2; with H=0 this is 2K=-|sigma|^2, and one must additionally use chi=0 and Gauss-Bonnet to obtain K=0 and |sigma|=0. This is a genuine omitted proof step, but it is fillable by standard arguments and is not circular, since Proposition 2.2 is proved independently and does not assume the rigidity conclusion. The sole self-citation, [21], is contextual and is not used as a step in any proof. Therefore the paper contains no significant circularity.

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

There are no fitted parameters, no hand-chosen constants, and no invented geometric or physical entities. The argument rests on standard spectral geometry, the Li-Yau conformal-area method, and two external classification theorems. The only ad hoc assumption is the unsupported implication in the Theorem B equality case.

assumptions (4)
  • standard math Li-Yau conformal center-of-mass lemma: for a positive density φ dv on an immersed surface in S^n there exists y in B^{n+1} with ∫ φ (F_y ∘ x) dv = 0.
    Invoked in Sections 3.3 and 4 to construct test functions orthogonal to the first eigenfunction of L; cited to Li-Yau [18] and Hersch [9].
  • standard math Min-max principle and positivity of the first eigenfunction for the Jacobi operator L.
    Defines λ2 and justifies using test functions in the Rayleigh quotient estimates throughout Theorems 3.3 and 4.1.
  • standard math El Soufi-Ilias classification of tori immersed by first eigenfunctions (Theorem 3.1) and El Soufi-Giacomini-Jazar classification of the Klein bottle (Theorem 3.2).
    These external classification results are used to finish the equality case of Theorem A.
  • ad hoc to paper Equality in Proposition 2.2 together with minimality implies flatness and total geodesy in S1(r) × S2(s).
    This is the unproved step in the proof of Theorem B. Proposition 2.2 only yields, for r = 1/√2, either a slice or a torus with 2K = 4|H|^2 - |σ|^2; with H = 0 this gives 2K = -|σ|^2, not flatness. The implication is load-bearing and is not established in the text.

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Pith. "Pith review of Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator." pith.science (2026). https://pith.science/paper/XXGDLHI6

@misc{pith2026250522439,
  author       = {Pith},
  title        = {Pith review of: Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXGDLHI6}},
  note         = {Machine review of arXiv:2505.22439}
}
abstract

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$.

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