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There exist positively curved metrics on the two-sphere such that the first eigenfunction has arbitrarily many non-degenerate local maxima.

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load-bearing objection The paper constructs metrics on the 2-sphere with positive curvature where the first eigenfunction has arbitrarily many non-degenerate local maxima, giving a direct negative answer to the Grossi-Provenzano open question.

arxiv 2606.26419 v1 pith:6PA7NSAO submitted 2026-06-24 math.DG

Arbitrarily Many Non-degenerate Local Maxima of First Nonzero Eigenfunctions on Positively Curved Two-spheres

Heng Zhang This is my paper
classification math.DG
keywords first eigenfunctionlocal maximapositive Gaussian curvaturetwo-sphereLaplace eigenvaluesimple eigenvalueRiemannian metricnon-degenerate critical points
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that for any integer m greater than or equal to 2, one can find a Riemannian metric on a surface diffeomorphic to the 2-sphere with strictly positive Gaussian curvature. In this metric the first nonzero eigenvalue of the Laplacian is simple. The corresponding eigenfunction, after normalization and sign choice, possesses at least m distinct non-degenerate local maxima. This result provides a negative answer to an open question, indicating that positive curvature does not restrict the number of peaks of the first eigenfunction. A reader would care because it shows that the lowest eigenfunction on a positively curved sphere can exhibit more complex critical point behavior than the round sphere exhibits.

Core claim

For every integer m ≥ 2, there exists a smooth closed Riemannian surface (M,g) diffeomorphic to S², with positive Gaussian curvature, such that λ₁(M) is simple and, after choosing the sign, its normalized first nonzero Laplace eigenfunction has at least m distinct non-degenerate local maxima.

What carries the argument

A metric construction on the 2-sphere that introduces additional non-degenerate local maxima into the first eigenfunction while preserving positive Gaussian curvature and simplicity of λ₁.

Load-bearing premise

The construction succeeds in adding arbitrarily many maxima without violating positive curvature or making the eigenvalue non-simple.

What would settle it

An explicit metric or theorem proving that for some m the first eigenfunction on every positive-curvature metric on S² has strictly fewer than m non-degenerate local maxima.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 1 minor

Summary. The manuscript proves that for every integer m ≥ 2 there exists a smooth closed Riemannian surface (M, g) diffeomorphic to S² with positive Gaussian curvature such that λ₁(M) is simple and, after sign choice, its normalized first nonzero eigenfunction possesses at least m distinct non-degenerate local maxima. This furnishes a negative answer to the open question posed by Grossi and Provenzano (Math. Ann. 389(4): 3447–3470, 2024).

Significance. If the existence construction is valid, the result is significant: it demonstrates that no uniform upper bound exists on the number of non-degenerate local maxima of the first eigenfunction when the metric is restricted only to positive Gaussian curvature and λ₁ simplicity. This enlarges the known range of possible critical-point configurations for λ₁-eigenfunctions on positively curved spheres and supplies a concrete counter-example to a natural expectation in the literature.

minor comments (1)
  1. The abstract is concise, but the manuscript should explicitly state in the introduction or §1 the precise mechanism (e.g., perturbation scheme, gluing construction, or curvature-control estimates) used to add arbitrarily many maxima while keeping K > 0 everywhere and λ₁ simple.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the main result and for noting its potential significance as a counterexample to the question of Grossi and Provenzano. No major comments were listed in the report, so we provide no point-by-point responses below. The recommendation of 'uncertain' appears to reflect a need for verification of the construction; the full details are in the manuscript, and we remain available to address any specific questions about the proof.

Circularity Check

0 steps flagged

No significant circularity; existence proof is self-contained

full rationale

The paper states a pure existence result: for every m≥2 there exists a positively curved metric on S² making λ₁ simple with the first eigenfunction having ≥m non-degenerate maxima. No equations, fitted parameters, or self-citations are exhibited that would reduce this claim to a tautology or to a prior result by the same authors. The abstract and available description supply no load-bearing steps that collapse by construction, so the derivation chain remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information is given on free parameters, background axioms, or new entities introduced in the construction.

pith-pipeline@v0.9.1-grok · 5617 in / 1167 out tokens · 32862 ms · 2026-06-26T00:53:42.294084+00:00 · methodology

0 comments
read the original abstract

We prove that, for every integer $m\ge 2$, there is a smooth closed Riemannian surface $(M,g)$ diffeomorphic to $\mathbb S^2$, with positive Gaussian curvature, such that $\lambda_1(M)$ is simple and, after choosing the sign, its normalized first nonzero Laplace eigenfunction has at least $m$ distinct non-degenerate local maxima. This gives a negative answer to the Open Question of Grossi and Provenzano (Math. Ann. 389(4): 3447--3470, 2024).

Figures

Figures reproduced from arXiv: 2606.26419 by Heng Zhang.

Figure 1
Figure 1. Figure 1: A schematic contrast related to Question 1.1. On the round sphere, the height function has exactly two critical points, a maximum and a minimum. In contrast, on suitably chosen positively curved spheres, a second eigenfunction may have several distinct non-degenerate local maxima. The main result of this paper gives a negative answer in a strong form. Positive Gaussian curvature alone does not force a seco… view at source ↗
Figure 2
Figure 2. Figure 2: The metric double X = DΩ of a convex polygon Ω. Two copies Ω + and Ω − are glued by the identity map along their common boundary ∂Ω. Lemma 3.1 (Sobolev structure of the double). Let Ω ⊂ R 2 be a bounded convex polygonal domain and let X = DΩ. Then, identifying a function on X with its restrictions to the two sheets, (3.2) H1,2 (X) =  (f+, f−) ∈ H1 (Ω) ⊕ H1 (Ω) : Tr f+ = Tr f− on ∂Ω [PITH_FULL_IMAGE:figur… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the pancake approximation. A convex polygon Ω is first replaced by smooth strictly convex inner domains Ωδ, and then by thin strictly convex surfaces Mε,δ = ∂Cε,δ with positive Gaussian curvature. Lemma 4.3. For every sufficiently small δ > 0 and every ε > 0, Mε,δ is a smooth embedded two-sphere with positive Gaussian curvature [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

discussion (0)

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Reference graph

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